Effective reconstruction of generic genus 5 curves from their theta hyperplanes

TL;DR

This paper presents a method to reconstruct generic genus 5 curves from their theta hyperplanes, providing a complete description of the Schottky locus in genus 5 through effective computational techniques.

Contribution

It introduces an effective reconstruction method for genus 5 curves from theta hyperplanes and describes the Schottky locus explicitly using this data.

Findings

Successful reconstruction of genus 5 curves from theta hyperplanes

Complete description of the Schottky locus in genus 5

Use of certified numerical algorithms for proof

Abstract

We effectively reconstruct the set of enveloping quadrics of a generic curve C of genus 5 from its theta hyperplanes; for a generic genus 5 curve C this data suffices to effectively reconstruct C. As a consequence we get a complete description of the Schottky locus in genus 5 in terms of theta hyperplanes. The computational part of the proof is a certified numerical argument.

Equations

$$\{\theta :2\theta =K_C\text{ and }\dim H^0(C,\theta )=\dim H^0(C,\theta +\alpha )\equiv 1\mathrm{mod}2\}.$$

$$I_2(C)=\bigcap _{\alpha \in JC[2]\setminus \{0\}}\mathrm{span}(\{q_{\{\theta ,\theta +\alpha \}}{\}}_{\theta \in {\Sigma }_{C,\alpha }})$$

$$p_D:H^0({\mathcal{O}}_{|D{|}^{\ast }}(2))\setminus \{q:q{|}_{{\varphi }_D(C)}=0\}$$

$$q$$

$$p_{K_C}^{-1}{p}_{K_C+\alpha }{H}^0({\mathcal{O}}_{|K_C+\alpha {|}^{\ast }}(2))\subset H^0({\mathcal{O}}_{|K_C{|}^{\ast }}(2)).$$

$$\dim ({\cap }_{i=1}^n{\mathcal{V}}_i{|}_x)$$

$$\dim \mathrm{span}(\{q_{\{\theta ,\theta +\alpha \}}{\}}_{\theta \in {\Sigma }_{C_0,\alpha }})=13,\text{ for all }\alpha \in A,$$

$$\dim {\cap }_{\alpha \in A}\mathrm{span}(\{q_{\{\theta ,\theta +\alpha \}}{\}}_{\theta \in {\Sigma }_{C_0,\alpha }})=3$$

$$V_{C,\alpha }\supset \mathrm{span}(\{q_{\{\theta ,\theta +\alpha \}}{\}}_{\theta \in {\Sigma }_{C,\alpha }}),I_2(C),$$

$$\dim V_{C,\alpha }\le \dim {\mathrm{Sym}}^2{H}^0(K_C+\alpha )+\dim I_2(C).$$

$$\dim$$

$$=$$

$$\chi ({\mathrm{Pic}}^2{E}_i)\cap {\Gamma }^{\prime }=\{[Q_{i+1}^{\prime }],[Q_{i-1}^{\prime }],[Q_{i-2}],[Q_{i+2}]\}.$$

$${\beta }_i:={\chi }^{-1}[Q_{i+2}]-{\chi }^{-1}[Q_{i-2}],{\alpha }_i:={\chi }^{-1}[Q_{i+2}]-{\chi }^{-1}[Q_{i-1}^{\prime }].$$

$${\chi }^{-1}[Q_{i-2}]-{\chi }^{-1}[Q_i^{\prime }]={\chi }^{-1}[Q_{i+1+2}]-{\chi }^{-1}[Q_{i+1-1}^{\prime }],$$

$${\chi }^{-1}[Q_{i+2}]-{\chi }^{-1}[Q_i^{\prime }]={\chi }^{-1}[Q_{i-1-2}]-{\chi }^{-1}[Q_{i-1+1}^{\prime }]={\chi }^{-1}[Q_{i-1+2}]-{\chi }^{-1}[Q_{i-1-1}^{\prime }],$$

$${\mathrm{Pic}}^{0}C[2]=\mathrm{span}({\alpha }_0,\dots ,{\alpha }_4)\oplus \mathrm{span}({\alpha }_0^{\prime },\dots {\alpha }_4^{\prime }),$$

$$2({\pi }_i^{-1}(q_{ij}^k)+{\pi }_j^{-1}(q_{ji}^{k^{\prime }}))={\pi }_i^{-1}(\overline{p_{ij}})+{\pi }_j^{-1}(\overline{p_{ji}})=2{\theta }_{ij}=K_C$$

$$V_{ij}$$

$$=$$

$$V_{ij}^{\perp }=\{\begin{array}{ll}\mathrm{span}\{{\alpha }_1,\dots {\alpha }_5\}\oplus \mathrm{span}({\alpha }_{i+3}^{\prime })& \text{for }j=i+1\\ \mathrm{span}\{{\alpha }_1,\dots {\alpha }_5\}\oplus \mathrm{span}({\alpha }_{i-1}^{\prime }+{\alpha }_{i+1}^{\prime }+{\alpha }_{i+3}^{\prime })& \text{for }j=i+2,\end{array}$$

$$JC[2]=\mathrm{span}\{{\alpha }_i{\}}_{i=1}^5\oplus \mathrm{span}\{{\alpha }_i^{\prime }{\}}_{i=1}^5.$$

$$\forall a\in \left(\genfrac{}{}{0ex}{}{O_{160}}{4}\right)\setminus A:\sum _{\theta \in a}\theta \ne 2K_C.$$

$${\gamma }_k^I:={\alpha }_k^{\prime }+{\alpha }_{k+1}^{\prime },{\gamma }_k^{II}:={\alpha }_k^{\prime }+{\alpha }_{k+2}^{\prime },{\gamma }_k^{III}:=(\sum _m{\alpha }_m^{\prime })-{\alpha }_k^{\prime }.$$

$${\alpha }_{i+3}^{\prime }+{\alpha }_{i^{\prime }+3}^{\prime }=\{\begin{array}{ll}{\gamma }_{i+3}^I& \text{for }{i}^{\prime }=i+1\\ {\gamma }_{i+3}^{II}& \text{for }{i}^{\prime }=i+2\end{array},$$

$${\alpha }_{i-1}^{\prime }+{\alpha }_{i+1}^{\prime }+{\alpha }_{i+3}^{\prime }+{\alpha }_{i^{\prime }-1}^{\prime }+{\alpha }_{i^{\prime }+1}^{\prime }+{\alpha }_{i^{\prime }+3}^{\prime }=\{\begin{array}{ll}{\gamma }_{i-1}^{III}& \text{for }{i}^{\prime }=i+1,\\ {\gamma }_{i-1}^I& \text{for }{i}^{\prime }=i+2,\end{array}$$

$${\alpha }_{i+3}^{\prime }+{\alpha }_{i^{\prime }-1}^{\prime }+{\alpha }_{i^{\prime }+1}^{\prime }+{\alpha }_{i^{\prime }+3}^{\prime }=\{\begin{array}{ll}{\gamma }_{i^{\prime }-1}^{II}& \text{for }{i}^{\prime }=i\\ {\gamma }_{i^{\prime }}^{III}& \text{for }{i}^{\prime }=i+1\\ {\gamma }_{i^{\prime }+3}^I& \text{for }{i}^{\prime }=i+2\\ {\gamma }_{i^{\prime }+2}^{III}& \text{for }{i}^{\prime }=i+3\\ {\gamma }_{i^{\prime }+1}^{II}& \text{for }{i}^{\prime }=i+4.\end{array}$$

$${\overline{\gamma }}_k^I:=$$

$${\overline{\gamma }}_k^{IV}:=$$

$$V_{i,i+1}=$$

$$=$$

$$V_{i,i+2}=$$

$$=$$

Full text

1. 1 Introduction
2. 2 Proving the general result by specialization
3. 3 Odd theta characteristics of Humbert curves
4. 4 Subsets of Steiner systems on Humbert curves
5. 5 The Wiman curve
6. 6 Certifying a divisor is not two-canonical
7. 7 Certified numeric proof of the main theorem

1. Introduction

The quest for methods of reconstructing a curve from its theta hyperplanes goes back to the 19th and early 20th century geometers Aronhold and Coble (see [A], [Co] chapter IV, and [D] sections 6.1.2 and 6.2.2): in the non hyperelliptic genus 3 case, theta hyperplanes are simply bitangents, and both Aronhold and Coble provided formulas for reconstructing curves from certain ordered subsets of the $28$ bitangents of the curve.

Recent years witnessed some revived interest in generalizations of this problem from several directions: first relaxing the need for ordered theta hyperplanes (see [CS1], [L1]) (where “order” means known action of the level-2 cover deck transformations group $SP_{2\mathrm{genus}}(2)$), and then generalizations to higher genus curves (see [CS2]) and Abelian varieties (see [GS-M1], [GS-M2]). Using the fact that [CS2] relaxed the need for ordered theta characteristic, we gave in [L2] two results for generic genus $4$ curves: An explicit construction of the enveloping quadric, and an explicit construction of the nodes of Wirtinger cubics, which [CKRN] used to construct the variety of enveloping cubics of $C$, thus completing the reconstruction of $C$.

The object of this paper is to extend — in Theorem 1.1 — the first of the two results from [L2] to genus $5$. Throughout this paper we consider a generic complex curve $C$ of genus $5$; since $C$ is generic we assume that all its odd theta characteristics are $1$ dimensional - i.e. if $\theta$ is an odd theta characteristic of $C$ then $\dim H^{0}(C,\theta)=1$. Hence, for each odd theta characteristic $\theta$ there exists a unique hyperplane $l_{\theta}$ — called a theta hyperplane — in the dual canonical system of $C$ such that when $C$ is identified with its canonical image, the points in the intersection product $C\cdot l_{\theta}$ are all double, and the points in $\frac{1}{2}C\cdot l_{\theta}$ sum up to $\theta$. This hyperplane is the projectivization of the plane $T_{\theta}\Theta_{C}\subset T_{\theta}JC$ under the identification of $T_{\theta}JC=T_{0}JC=H^{0}(K_{C})^{*}$, and where $JC,\Theta_{C}$ are the Jacobian variety and theta divisor of the curve $C$. Recall (see e.g. [D] 5.4.2), that if $\alpha$ is a non-trivial $2$ torsion point on the Jacobian $JC$, then the Steiner system $\Sigma_{C,\alpha}$ of the pair $(C,\alpha)$ is defined to be the set

[TABLE]

The number of theta characteristics in a Steiner system of a genus $g$ curve is $2^{g-1}\cdot(2^{g-1}-1)$; i.e. in our case a Steiner system is comprised of $16\cdot 15=240$ odd theta characteristics, out of the total of $2^{5-1}(2^{5}-1)=496$ odd theta characteristics of the curve. For each pair $\theta,\theta+\alpha$, and corresponding theta hyperplanes $l_{\theta},l_{\theta+\alpha}$, we let $q_{\{\theta,\theta+\alpha\}}\in|{\mathcal{O}}_{|K_{C}|^{*}}(2)|$ be the image of $\{l_{\theta},l_{\theta+\alpha}\}\in S^{2}|K_{C}|$ under the map: $S^{2}|K_{C}|\to|{\mathcal{O}}_{|K_{C}|^{*}}(2)|$. We can now state the following:

Theorem 1.1**.**

Let $C$ be a generic genus $5$ curve, with $q_{\theta,\theta+\alpha}$ as above, then the set of quadric enveloping $C$ is given by:

[TABLE]

We note two immediate corollaries of this theorem:

Corollary 1.2**.**

For a generic genus $5$ curve $C$, Theorem 1.1 effectively determines the curve.

Proof.

By the Enriques-Babbage Theorem (see e.g. [ACGH] chapter VI §3), a generic canonical curve is either trigonal, isomorphic to a plane quintic, or is cut out by quadrics, in which case $C=\cap_{[q]\in I_{2}(C)}Z(q)$. ∎

Since theta hyperplanes are defined for any principally polarized Abelian variety $(A,\Theta)$ via the Gauss map, we may define $\mathbb{I}_{2}(A,\Theta)$ as the right hand side of the equation in Theorem 1.1. Using this definition we get:

Corollary 1.3**.**

The Schottky locus in genus $5$ is comprised of (the closure of) pairs $(A,\Theta_{A})$ such that the intersection $\cap_{[q]\in\mathbb{I}_{2}}Z(q)$ is a curve $C$ of arithmetic genus $5$, and such that the moduli points $[(JC,\Theta_{C})],[(A,\Theta)]$ in the moduli of principally polarized Abelian varieties are identical.

Proof.

This follows from Theorem 1.1 and the previous corollary. ∎

The paper is organized as follows: In Section 2 we show how to reduce the proof of the theorem to slightly stronger claims on a specific curve: as in [L2] most of the proof is done by specializing to one curve; in the case at hand the Wiman curve ${{W}}=W^{160}$.

Sections 3, 4 and 5 form the bulk of the algebro-geometric part of the paper. In Section 3 we recall the definition of Humbert curves, and concretely describe a set of $160$ theta characteristics on them. In the remainder of the paper we carefully analyze this set of odd theta characteristics. In Section 4 we give a partial description of the intersections of pairs of points in the set of $160$ theta characteristics we found with the Steiner systems of Humbert curves: we show that these pairs intersect exactly $510$ out of the $1023$ Steiner systems, and compute the sizes of the intersections. In Section 5 we introduce the Wiman curve as a special Humbert curve with a large automorphism group. We compute explicit coordinates for the theta characteristics we found in Section 3, and decompose the second symmetric tensor of the canonical system of ${{W}}$ to irreducible representation of its automorphism group. The most immediate gain of the later is a natural decomposition of the space of quadrics in the canonical system to a direct sum of $I_{2}({{W}})$ and $H^{0}(2K_{{W}})$.

In Section 6 we show how to produce a certified numerical proof that a given sum of four theta characteristics on ${{W}}$ is not 2-canonical. In a nutshell, we show how to certify that all the non-trivial quadrics in $H^{0}(2K_{{W}})$ — considered as the orthogonal complement of $I_{2}({{W}}))$ — are bounded away from zero on the sum of the theta characteristics at hand. We note that while certified numerical arguments are not new to algebraic geometry (see e. g. [BST], [HS], and the bertini package), they usually deal with a harder case of assuring the existence of a true zero in the neighborhood of an approximately computed one. The approach we take is closer in spirit to e.g. [OS]; we simply bound values away from zero.

Finally, in Section 7 we connect the results from Section 6 with actual numerical error bounds, tie it with the computation of Section 4 to completely describe the intersections of pairs of theta characteristics found in Section 3 with all Steiner systems; then use the decomposition into irreducible representations from Section 5 to produce a certificate for the numerical computation of the dimension of the intersection in Theorem 1.1. We also provide a witness (in the proof theoretic sense), which can be verified by a relatively small number of “traditional” matrix rank computation over a bi-quadratic extension of the rational numbers. It is important to note that the computer program accompanying this paper, and described in Section 7 is an integral part of this paper. This program can be found at https://arxiv.org/src/1908.02355v2/anc/W160_steiner_systems_and_IC2.cc.

Sadly, unlike the theorem itself, which I conjecture generalizes to higher genera, it is hard to see how the proof technique from this paper and from [L2] will generalize to genera greater than $14$, where one cannot work directly with a “general curve”. In the “next” cases, genera 6,7, the obvious candidates to work with are curves with large automorphism groups: e.g. some curve along the Wiman-Edge pencil (see [DFL]) in genus 6, and the Fricke-Macbeath curve in genus 7 (see [M]). We note the genus 6 case is especially interesting for two reasons: it is the first genus where the general curve is not a complete intersection, and it is the first case where the Schottky problem is not fully resolved.

Acknowledgments

I would like to thank T. Celik and A. Kulkarni for many useful remarks they gave on an early draft of this paper.

2. Proving the general result by specialization

Notation 2.1**.**

Given a curve $C$ and a divisor $D\in\mathrm{Div}C$, denote by $\phi_{D}$ the map from $C$ to the dual linear system $|D|^{*}$; we identify $C$ with it’s canonical image $\phi_{K_{C}}(C)$. Denote by $\alpha$ some point in $JC[2]\smallsetminus\{0\}$. Denote the null set of a homogeneous polynomial $f$ by $Z(f)$. Finally denote the projectivization of a space $V$ by ${\mathbb{P}}V$.

Definition 2.2** (Steiner spaces).**

We set

[TABLE]

Note that there are natural isomorphisms $H^{0}({\mathcal{O}}_{|D|^{*}}(2))\cong\mathrm{Sym}^{2}H^{0}(D)$. With $C,\alpha$ as above, denote by $V_{C,\alpha}$ the closure of the pre-image

[TABLE]

The authors of [CKRN] suggested to name $V_{C,\alpha}$ Steiner spaces.

We want to reduce Theorem 1.1, which is stated for a generic curve, to a somewhat stronger claim about a single specific curve. To this end we use a standard degeneration argument:

Lemma 2.3** (see e.g. Corollary 7 from [L2]).**

Let ${\mathcal{V}}/X$ be a vector bundle over a base $X$, and let ${\mathcal{V}}_{1},\dots\mathcal{V}_{n}$ be sub-bundles of ${\mathcal{V}}$. Then the function $\dim\langle{\mathcal{V}}_{1}|_{x},\ldots,{\mathcal{V}}_{n}|_{x}\rangle$ is lower semi-continuous on $X$, and the function

[TABLE]

is upper semi-continuous on $X$.

Armed with the lemma we now prove the following reduction:

Proposition 2.4**.**

To prove Theorem 1.1 it suffices to exhibit one non-hyperelliptic curve $C_{0}$ and a set $A\subset JC_{0}[2]\smallsetminus\{0\}$ so that

[TABLE]

and

[TABLE]

Proof.

It is clear from Definition 2.2 that

[TABLE]

and that

[TABLE]

Thus, to prove that $V_{C,\alpha}$ and $\mathrm{span}(\{q_{\{\theta,\theta+\alpha\}}\}_{\theta\in\Sigma_{C,\alpha}})$ are equal it suffices to show that for the curve $C_{0}$:

[TABLE]

The claim now follows from Lemma 2.3 ∎

3. Odd theta characteristics of Humbert curves

Proposition-Definition 3.1** (Humbert’s curves, originally defined in [H]).**

Assume $C$ is a non hyperelliptic genus $5$ curve, then $C$ sits on a smooth quadric if and only if is it is non trigonal (see [ACGH], chapter VI exercises F1, F2). Henceforth we further assume that $C$ is also non-trigonal. Denote by $\Gamma\subset{\mathbb{P}}I_{2}(C)$ the locus of quadrics of rank $\leq 4$, and by $\Gamma^{\prime}$ the locus of quadric of rank $\leq 3$. Denote by $W^{r}_{d}(C)\subset\mathrm{Pic}^{d}(C)$ the subscheme of degree $d$ divisors $D$ so that $\dim|D|\geq r$. Then by [ACGH] chapter VI exercise batch F, the following properties hold (for proofs see the specific exercises in each parenthesis):

• •

$\Gamma$ is a plane quintic with no multiple components and at most ordinary nodes, which are exactly $\Gamma^{\prime}$ (see exe. 4).

• •

The map $\chi:W^{1}_{4}(C)\to\Gamma$ defined by $D\mapsto{\mathbb{P}}(\text{Tangent cone}_{D}W^{0}_{4}(C))$ is a double cover, branched over $\Gamma^{\prime}$. Specifically, $W^{1}_{4}(C)_{\text{sing}}=\{D\in W^{1}_{4}(C):2D=K_{C}\}$ (see exe. 5, 6, 7).

• •

$W^{1}_{4}(C)$ is irreducible if and only if $\Gamma$ is (see exe. 10).

• •

If $C\to E$ is a bielliptic double cover, then there is a component $\Sigma$ of $W^{1}_{4}(C)$ corresponding to the $g^{1}_{2}$s on $E$. These components project to line components on $\Gamma$, and vice-versa. (see exe. 11, 12).

• •

$C$ has five bi-elliptic pencils if and only if it has 10 semi-canonical pencils, if and only if, after a suitable change of coordinates — called diagonalization — ${\mathbb{P}}I_{2}(C)$ is spanned by three quadrics of the form $Q_{\alpha}=\sum_{i=0}^{4}\lambda_{\alpha,i}x_{i}^{2}$. (see exe. 13, 14). These are called Humbert curves.

Collecting the properties above we see that on a Humbert curve the variety $W^{1}_{4}(C)$ consists of five subvarieties $\mathrm{Pic}^{2}(E_{i})$ — where each $E_{i}$ is an elliptic double quotient of $C$ — which intersect at $\binom{5}{2}=10$ points. Each of these five $\mathrm{Pic}^{2}(E_{i})$ is a double cover of one line in $\Gamma$ under the $\chi$.

For an alternative presentation to the one above, see either [H], [E] or [V].

Proposition-Definition 3.2**.**

Let $C$ be a Humbert curve, and assume diagonalized coordinates $x_{0},\ldots,x_{4}$ on the canonical system. By elimination on the monomials $x_{0}^{2},\ldots,x_{4}^{2}$ there are exactly $\binom{5}{3}=10$ quadrics (up to constants) in $I_{2}(C)$ supported on only three coordinates. Picking any four coordinates, four of these 10 quadrics are supported on a subset of the four coordinates. By definition, the points corresponding to these four quadrics in ${\mathbb{P}}I_{2}(C)$ sit on a line, and the intersection of their null sets is a cone over an elliptic curve (the intersection of the quadrics in 3-dimensional projective space) in the canonical system of $C$. Thus we have identified the elliptic double quotients $E_{i}$; we fix the indexing so that $i$ is the vanishing coordinate of the cone over $E_{i}$. As for the realizing the semi-canonical pencils, each of the 10 quadrics supported on 3 coordinates is a cone over some conic in the ${\mathbb{P}}^{2}$ generated by these coordinates.

Taking indices modulo $5$, denote the semi-canonical pencil of the form $ax_{i-1}^{2}+bx_{i}^{2}+cx_{i+1}^{2}$ by $Q_{i}$, and the semi-canonical pencil of the form $ax_{i-2}^{2}+bx_{i}^{2}+cx_{i+2}^{2}$ by $Q_{i}^{\prime}$. Denote by $\pi_{i}$ the five double elliptic covers $C\to E_{i}$. We will identify $\mathrm{Pic}^{2}E_{i}$ with its image in $W^{1}_{4}(C)$. Denote the points representing $Q_{i},Q_{i}^{\prime}$ on $\Gamma^{\prime}\subset\Gamma\subset I_{2}(C)$ by $[Q_{i}],[Q_{i}^{\prime}]$. Using these notations we have

[TABLE]

Denote

[TABLE]

Henceforth, we will always assume our Humbert curves are diagonalized, with $E_{i},Q_{i},Q_{i}^{\prime}$ as in Proposition-Definition 3.2.

Proposition 3.3**.**

Consider $\alpha_{i},\beta_{i}$ as degree 0 divisors on $\mathrm{Pic}^{2}E_{i}\subset W^{1}_{4}(C)$, then any curve isomorphism $\mathrm{Pic}^{2}E_{i}\to E_{i}$, induces the same natural isomorphism $\mathrm{Pic}^{0}(\mathrm{Pic}^{2}E)=\mathrm{Pic}^{0}E$. Moreover, this isomorphism takes $\alpha_{i},\beta_{i}$ to distinct non-trivial points in $\mathrm{Pic}E_{i}[2]$

Proof.

The first claim follows since for any elliptic curve we have $\mathrm{Pic}^{0}E\cong E$ as curves. As for the second claim, they are differences between one ramification point under $\chi$, and two other ramification points. ∎

Notation 3.4**.**

Henceforth we identify $\alpha_{i},\beta_{i}$ with their respective images in $\mathrm{Pic}^{0}(E_{i})$.

Proposition 3.5** (The kernel of the map $\prod E_{i}\to JC$).**

The map $\prod JE_{i}\to JC$ is a degree $2^{5}$ isogeny of principally polarized Abelian varieties, whose kernel is generated by $\{\beta_{i}-\alpha_{i+1}-\alpha_{i-1}\}_{i=1}^{5}$, where indices are taken mod $5$.

Proof.

The inverse image of the point $(e_{0},e_{1},e_{2},e_{3},e_{4})\in\prod\mathrm{Pic}^{1}E_{i}$ under the map $\prod_{i}\pi_{i}:\mathrm{Pic}C\to\prod_{i}\mathrm{Pic}E_{i}$ is $\{(c_{0},\ldots,c_{4})|c_{i}\in\pi^{-1}e_{i}\}$. Hence the map $JC\to\prod JE_{i}$ is of degree $2^{5}$, hence so is the dual map $\prod JE_{i}\to JC$. We now consider, for a given $i$, the points $\chi^{-1}[Q_{i+2}],\chi^{-1}[Q_{i-2}],\chi^{-1}[Q^{\prime}_{i}]$. Computing on $E_{i}$, we see that the differences between the first two points is the image of $\beta_{i}$ under the map $\prod\mathrm{Pic}E_{i}\to\mathrm{Pic}C$. Computing on $E_{i+1}$, the difference between the second and the third is

[TABLE]

which is the image of $\alpha_{i+1}$. Finally computing on $E_{i-1}$ the difference between the third and the first is the image of

[TABLE]

which is the image of $\alpha_{i-1}$, and where the last equality is by passing from two branch points of a $g^{1}_{2}$ on an elliptic curve to the two complimentary ones. Hence, the image of $\beta_{i}+\alpha_{i+1}+\alpha_{i-1}$ under $\prod\mathrm{Pic}E_{i}\to\mathrm{Pic}C$ is trivial; since $\beta_{i}$ is a 2-torsion point, this statement is equivalent to the claim in the Proposition. It remains to show that the span of the $\alpha_{i}$ is an isotropic group w.r.t. the Weil pairing, which is clear since they are supported on different components in the product $\Pi E_{i}$. ∎

Notation 3.6**.**

We identify the $\alpha_{i}$ with their images in $\mathrm{Pic}^{0}C[2]$, and denote by $\alpha_{i}^{\prime}$ the unique element whose Weil pairing with $\alpha_{j}$ is non-trivial if and only if $j=j^{\prime}$. I.e.

[TABLE]

where the only non-trivial Weil pairings between the ten basis elements $\alpha_{i},\alpha^{\prime}$ are the five pairings $\langle\alpha_{i},\alpha^{\prime}_{i}\rangle=1$.

Proposition-Definition 3.7** (“easy to describe” effective theta characteristics).**

Let $E_{i},E_{j}$ be such that $\chi(\mathrm{Pic}^{2}E_{i})\cap\chi(\mathrm{Pic}^{2}E_{j})=[Q_{l}]$, then the $\chi$ pullback of $[Q_{l}]$ is the intersection point of $\mathrm{Pic}^{2}E_{i}\cap\mathrm{Pic}^{2}E_{j}$ with multiplicity $2$. Denote this intersection point by $P$, and its pullbacks to $E_{i},E_{j}$ by $p_{ij},p_{ji}$ respectively. Denote by $q_{ij}^{k}$ (where $k=1,\ldots,4$) the points on $E_{i}=\mathrm{Pic}^{1}E_{i}$ so that $2q_{ij}^{k}=p_{ij}$; similarly denote by $q_{ji}^{k^{\prime}}$ the points on $E_{j}=\mathrm{Pic}^{1}E_{j}$ so that $2q_{ji}^{k^{\prime}}=p_{ji}$. Then:

[TABLE]

Hence $(\pi_{i}^{-1}(q_{ij}^{k})+\pi_{j}^{-1}(q_{ji}^{k^{\prime}}))$ is an effective theta characteristic.

Corollary 3.8**.**

The theta characteristics constructed above are odd and distinct (hence there are $10\times 2^{4}$ of them).

Proof.

It is clear that for each $\{i,j\}$ we get $2^{4}$ different effective theta characteristics, and that they are disjoint from the ten known effective even theta characteristics, which are the only effective even theta characteristics. Hence these $16$ theta characteristics are all odd. It remains to prove that the ten $16$-tuples are disjoint.

Note that there is no point $p\in C$ with two vanishing coordinates: Indeed if there was such a $p$ then, using the fact that each of the $Q_{i}$s and $Q^{\prime}_{i}$s are each supported on three coordinates, all the coordinates of $p$ would have to vanish.

Note also that of the four points on $C$ comprising the theta characteristic in Proposition-Definition 3.7, the two points in the $\pi_{j}$ pre-image of $q_{ij}^{k}$ have vanishing $i$ coordinate, their $j$ coordinate are interchangeable by $x_{j}\mapsto-x_{j}$, and their other coordinates are identical. Which means that — by the claim above — the theta $16$ tuple we get for a given $\{i,j\}$ is disjoint from the $16$ tuple we get from a different choice $\{i^{\prime},j^{\prime}\}$. ∎

Proposition-Definition 3.9**.**

By construction, each of the ten $16$-tuples of theta characteristics above is a translate of one of the spaces we now define:

[TABLE]

In the case where $|i-j|=1$, in which case w.l.o.g. $i+1=j$, the last span is $span(\alpha_{i},\alpha_{i-1},\alpha_{i+1},\alpha_{i+2}).$ In the second case w.l.o.g. $i+2=j$ in which case the span is $span(\alpha_{i},\alpha_{i-1}+\alpha_{i+1},\alpha_{i+2},\alpha_{i+1}+\alpha_{i+3})$.

Corollary 3.10**.**

Recalling Notation 3.6, and taking indices mod 5, the following identities hold in $JC[2]$:

[TABLE]

where orthogonality is w.r.t. the Weil pairing.

Proof.

The orthogonality to $\mathrm{span}\{\alpha_{1},\ldots\alpha_{5}\}$ is immediate. The orthogonality of the basis elements to the last element in the respective bases is by direct verification. ∎

Notation 3.11**.**

For each $i,j$ denote by $\eta_{ij}$ the unique non-trivial element in the projection of $V_{ij}^{\perp}$ on the second component in the direct sum decomposition

[TABLE]

4. Subsets of Steiner systems on Humbert curves

We refer the reader to Chapter 5 of [D] for background material on theta characteristics. Throughout this section we continue to work with a Humbert curve $C$.

Notation 4.1**.**

Denote by $O_{160}$ the set of 160 odd theta characteristics we found in Proposition-Definition 3.7. Denote by $O_{ij}$ the unique translate of $V_{ij}$ into $O_{160}$. We will call $\{\{\theta,\theta+\alpha\}|\theta\in\Sigma_{\alpha}\}$ the set of pairs of the Steiner system $\Sigma_{\alpha}$.

The main objective of this section is to prove the following:

Proposition 4.2**.**

Assume that $A\subset{O_{160}\choose 4}$ is such that

[TABLE]

Let $R$ be the symmetric relation on ${O_{160}\choose 2}$ defined by $\theta_{1}R\theta_{2}$ if $\theta_{1}\cup\theta_{2}\in A$. Further assume that $R$ is transitive and that $R$ partitions $O_{160}\choose 2$ into $510$ equivalent classes, then these equivalent classes are the intersections of $O_{160}\choose 2$ with the sets of pairs of the Steiner systems of $C$.

For “auxiliary” objectives of this section, and their motivation, see the discussion at the end of this section.

Recall (see e.g. [D] 5.1 for a detailed overview) that for any curve $C$, the set $JC[2]$ is a $2g$-dimensional symplectic space over ${\mathbb{F}}_{2}$ w.r.t. the Weil pairing, and that the set of theta characteristics of $C$, together with $\theta\mapsto h^{0}(\theta)\mod 2$, and with the addition action of $JC[2]$ on theta characteristics, is a corresponding affine symplectic space. Further recall that a subspace of a linear symplectic space $L\subset V$ is called isotropic if the symplectic pairing is trivial on it, and that in this case the symplectic pairing on $L$ induces a symplectic pairing on $L^{\perp}/L$.

Proposition 4.3**.**

Let $L$ be a $(g-1)$-dimensional isotropic subspace in a $2g$ dimensional linear symplectic space over ${\mathbb{F}}_{2}$, then $L$ has a unique translation in the odd points of a $2g$ affine symplectic spaces.

Proof.

Since $L$ is isotropic, it induces a symplectic pairing on the 2-dimensional (over ${\mathbb{F}}_{2}$) symplectic space $L^{\perp}/L$. Moreover, there is a one to to one correspondence between such translated spaces and the odd points of an affine space over the 2-dimensional space $L^{\perp}/L$, of which there is exactly one. ∎

Proposition 4.4**.**

Let $L$ be a maximal isotropic subspace of a $2g$ linear symplectic space over ${\mathbb{F}}_{2}$; let $L_{1},L_{2}$ be two distinct $g-1$ dimensional subspaces of $L$, let $l_{1},l_{2}$ be the unique non-trivial elements in the images of $L_{1},L_{2}$ in the 4 dimensional symplectic space $(L_{1}\cap L_{2})^{\perp}/(L_{1}\cap L_{2})$, and let $l^{\prime}_{1},l^{\prime}_{2}$ be so that $l_{1}^{\prime}\cdot l_{2}=0,l_{1}^{\prime}\cdot l_{1}=1,l_{2}^{\prime}\cdot l_{2}=1,l_{2}^{\prime}\cdot l_{1}=0$, where $\cdot$ is the symplectic pairing induced from the symplectic pairing on the original $2g$ dimensional space. Finally, let $O_{1},O_{2}$ be the unique translates (per Proposition 4.3) of $L_{1},L_{2}$ respectively into some affine $2g$ dimensional symplectic space over ${\mathbb{F}}_{2}$. Then $\{x-y|x\in O_{1},y\in O_{2}\}$ is the translate of $L$ by $l^{\prime}_{1}+l^{\prime}_{2}$. Moreover, each such difference is realized an equal number of times.

Proof.

We endow the space $(L_{1}\cap L_{2})^{\perp}/(L_{1}\cap L_{2})$ with an affine symplectic structure given by $\epsilon_{1}l_{1}+\epsilon^{\prime}_{1}l^{\prime}_{1}+\epsilon_{2}l_{2}+\epsilon^{\prime}_{2}l^{\prime}_{2}\mapsto\epsilon_{1}\epsilon_{1}^{\prime}+\epsilon_{2}\epsilon^{\prime}_{2}$. This induces affine structures on $L_{1}^{\perp}/L_{1}$ and $L_{2}^{\perp}/L_{2}$. The pre-image of the unique odd point of the 2 dimensional symplectic space $L_{1}^{\perp}/L_{1}$ (resp. $L_{2}^{\perp}/L_{2}$) in the 4 dimensional symplectic space $(L_{1}\cap L_{2})^{\perp}/(L_{1}\cap L_{2})$ under the natural projection is the pair $\{l_{1}+l_{1}^{\prime}+l_{2},l_{1}+l_{1}^{\prime}\}$ (resp. $\{l_{2}+l_{2}^{\prime}+l_{1}^{\prime},l_{2}+l_{2}^{\prime}\}$). Hence, the set of differences of these two pairs (here we compute the differences between points in an affine space, which is a linear space torsor, yielding a point in the linear space) is the translate by $l^{\prime}_{1}+l^{\prime}_{2}$ of the set $\{0,l_{1},l_{2},l_{1}+l_{2}\}$, which is the image of $L$ in the quotient space $(L_{1}\cap L_{2})^{\perp}/(L_{1}\cap L_{2})$ under the quotient map. Since all affine symplectic spaces of a given dimension are isomorphic, the claim about the identity of the set of differences follows. The claim about the number of representative for each difference being equal follows from the fact that the computations above were done in a quotient vector space. ∎

Proposition 4.5**.**

For each non zero element $\mu\in\mathrm{span}\{\alpha^{\prime}_{i}\}_{i=0}^{4}\cap(\sum_{i=0}^{4}\alpha_{i})^{\perp}$ there are exactly three pairs of indices $\{i,j\}\neq\{i^{\prime},j^{\prime}\}$ so that the difference set $O_{ij}-O_{i^{\prime}j^{\prime}}$ equals to $\mathrm{span}\{\alpha_{i}\}_{i=0}^{4}+\mu$, where each element appears 8 times for this difference. Moreover, this description covers all the possible indices choices $\{i,j\}\neq\{i^{\prime},j^{\prime}\}$.

Proof.

We work with indices mod 5. We start by naming the elements in $\mathrm{span}\{\alpha^{\prime}_{i}\}_{i=1}^{5}\cap(\sum_{i=0}^{4}\alpha_{i})^{\perp}\smallsetminus\{0\}$:

[TABLE]

By Proposition 4.4 the difference set $O_{ij}-O_{i^{\prime}j^{\prime}}$ equals to the translate of $\mathrm{span}\{\alpha_{i}\}_{i=1}^{5}$ by $\eta_{ij}+\eta_{i^{\prime}j^{\prime}}$, where each element in the translated span appears 8 times as a difference. Recalling the definition of the $\eta_{ij}$s in Notation 3.11, the sum $\eta_{ij}+\eta_{i^{\prime}j^{\prime}}$ equals:

• •

if $j=i+1,j^{\prime}=i^{\prime}+1$ (and w.l.o.g. $i^{\prime}-i\in\{1,2\}\mod 5$),

[TABLE]

• •

if $j=i+2,j^{\prime}=i^{\prime}+2$, (and w.l.o.g. $i^{\prime}-i\in\{1,2\}\mod 5$),

[TABLE]

• •

if $j=i+1,j^{\prime}=i^{\prime}+2$,

[TABLE]

The claim about each $\mu$ appearing $3$ times follows by counting. The claim about the multiplicity was already addressed, and the last claim is clear since $\binom{10}{2}=45=3\cdot(16-1)$. ∎

Proposition 4.6**.**

The difference set $O_{ij}-O_{ij}$ is $V_{ij}$, where each difference is represented $\#V_{ij}$ times; hence the difference set in $O_{ij}\choose 2$ is $V_{ij}\smallsetminus\{0\}$, where each difference is represented $\#V_{ij}/2=16/2=8$ times.

Proof.

The first part holds since $O_{ij}-O_{ij}=V_{ij}-V_{ij}$, and $V_{ij}$ is a vector space; the second part follows from symmetry. ∎

Proposition 4.7**.**

For any $\mu\in\mathrm{span}\{\alpha_{i}\}_{i=1}^{5}\smallsetminus\{0,\sum_{i=0}^{4}\alpha_{i}\}$, $\mu$ appears as a difference in one of the ten sets $(O_{ij}-O_{ij})\smallsetminus\{0\}$ precisely 48 times if $\mu$ is of the form $\alpha_{k},\alpha_{k}+\alpha_{k+2}$ or $\alpha_{k+2}+\alpha_{k+3}+\alpha_{k+4}$, and precisely 32 times otherwise, where we take indices modulo 5.

Proof.

We start by naming the non $\alpha_{i}$ elements in $\mathrm{span}\{\alpha_{i}\}_{i=1}^{5}\smallsetminus\{0,\sum_{i=0}^{4}\alpha_{i}\}$:

[TABLE]

Then the following equalities hold by direct verification:

[TABLE]

Observe that the $\alpha_{\bullet},\overline{\gamma}^{II}_{\bullet},\overline{\gamma}^{IV}_{\bullet}$s appears six times above, whereas the $\overline{\gamma}^{I}_{\bullet},\overline{\gamma}^{III}_{\bullet},\overline{\gamma}^{V}_{\bullet}$s appear four times. This means that when we consider the orbit of the above sets under coordinate rotation, we will have $30$ occurrences of elements of the first three types, and $20$ of the second. However, as each class of element is also of size five, and as rotations act transitively on each class, we see that each of the elements in the first classes is represented six times, and each of the elements in the later classes is represented four times. The result now follows from Proposition 4.6. ∎

Remark 4.8**.**

The orbit of each of the $\mu$’s from Proposition 4.7 under either the automorphism group of $C$, or its index $2$ subgroup generated by the rotations and sign changes on the coordinates, is of size 5.

Proof.

These groups act on the $\alpha_{k}$ via the action of dihedral group and the cyclic group on the indices, respectively. ∎

Proof of Proposition 4.2.

We start by noting that the set of differences between pairs of distinct elements of $O_{160}$ is $(\sum_{i}\alpha_{i})^{\perp}\smallsetminus\{0,\sum_{i}\alpha_{i}\}$: Indeed, in Propositions 4.5, 4.6 we showed that

[TABLE]

whereas in 4.7 we showed that

[TABLE]

By the definition of $A$, no members of two equivalence classes of $R$ are in the same set of pairs of some Steiner system. Moreover, since $O_{160}\choose 2$ has representatives in sets of pairs of $510$ Steiner systems, none of the equivalent classes can break into (sets of pairs of) Steiner systems. ∎

Sadly, this is as far as I managed to proceed with the non-numerical computation of the Steiner systems. It is important to note that the requirement for the certification of the numerical part is merely the number of partial Steiner systems represented in $\binom{O_{160}}{2}$, which is somewhat weaker than the enumerative parts of Propositions 4.5 and 4.6; we use the later ones, as well as Remark 4.8, for verification purposes (see 7.9).

5. The Wiman curve

Proposition-Definition 5.1** (Wiman curve ${{W}}=W^{160}$).**

In [W], Wiman discovered a genus $5$ curve (denoted in [E] by $W^{160}$) whose canonical model is the intersection of the nulls of the quadric forms:

[TABLE]

where $\zeta^{5}=1,\zeta\neq 1$. The automorphism group of this curve is the semi-direct product of $({\mathbb{Z}}/2)^{5}/({\mathbb{Z}}/2)$, acting by multiplying by $-1$ each of the coordinates, and the dihedral group $D_{5}$ acting on the coordinates, generated by $(x_{0},x_{1},x_{2},x_{3},x_{4})$, and $(x_{0},x_{3})(x_{1},x_{2})$. From now on we assume that the canonical system of the Wiman curve is endowed with these coordinates.

As explained in the introduction, the object of this section is two-fold: we explicitly compute the coordinates of $O_{ij}$ for the curve $W$, and we decompose the quadrics forms on the dual canonical system of $W$ to irreducible representations of the automorphism group.

Proposition-Definition 5.2** (The even effective theta characteristics of ${{W}}$).**

Denote by $\phi$ the “golden ratio” $(1+\sqrt{5})/2$. W.l.o.g.  we can assume that $\zeta=\exp(2\pi i/5)$, in which case

[TABLE]

We now compute

[TABLE]

where the last equality follows since $\phi-3=\frac{1}{\phi}-2=\frac{1-2\phi}{\phi}$. One gets the other $8$ even effective theta characteristics $Q_{i},Q_{i}^{\prime}$ by applying the cyclic group (which is a subgroup of the symmetry group of the curve) on the coordinates.

Proposition 5.3**.**

Identify the curve ${{W}}$ with its canonical image, and denote $a:=1/\sqrt{\phi},\ {\mathfrak{i}}:=\sqrt{-1}$. Then the effective theta characteristics from 3.7 are the orbit under the cyclic action ${\mathbb{Z}}/5{\mathbb{Z}}$ on the coordinates, (as a subgroup of the automorphism group of the curve) of the $32$ divisors:

[TABLE]

where $\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4}\in\{-1,1\}$ are sign choices.

Proof.

We find the first type of point by working (in the terminology of Proposition 3.7) with $i,j=2,3$; i.e. the construction corresponding to starting with the even theta characteristic corresponding to $Q_{0}$. Recall (see Proposition-Definition 3.2) that $E_{3}$ is the base of the cone $Z(Q_{0})\cap Z(Q_{4}^{\prime})$, whereas $E_{2}$ is the base of the cone $Z(Q_{0})\cap Z(Q_{1}^{\prime})$. We proceed to find $q_{32}^{k}$ (and later $q_{23}^{k}$). We start by finding the ramification points of the map from $E_{3}$ to the base of the cone $Z(Q_{0})$:

Assuming $x_{0}=0$ we get

[TABLE]

substituting in the equation of $Q_{4}^{\prime}$ we get $0=\phi x_{1}^{2}+\phi x_{2}^{2}+x_{1}^{2}$, and the ramification points of the map from $E_{3}$ to the base of the cone $Z(Q_{0})$ are the locus of the trivial $x_{2}$ discriminant; however the discriminant for $x_{2}$ is [math] if and only if $x_{1}=x_{2}=0$, which is impossible as at least one of the coordinates has to be non zero.

Hence $x_{0}\neq 0$. Since we work with homogeneous equations, we may choose $x_{0}\neq 0$ arbitrarily. Setting $x_{0}^{2}=-1/\phi$ and substituting in $Q_{0}$ we get:

[TABLE]

where the discriminant for $x_{2}$ is [math] when $x_{2}=0$, and then

[TABLE]

To find the pullbacks of these points to $C$ we solve $x_{3}$ in $Q_{1}^{\prime}$, giving us

[TABLE]

The coordinates written in the statement of the Proposition are projectively equivalent. To find $q_{23}^{k}$ we apply the symmetry switching between $Q_{4}^{\prime}$ and $Q_{0}^{\prime}$; i.e. switching $x_{1},x_{4}$ and $x_{2},x_{3}$.

As for the other type we consider $i,j=3,0$ (i.e. the construction corresponding to starting with the theta characteristic corresponding to $Q_{4}^{\prime}$): We now find the ramification points of the map from $E_{3}$ to the base of the cone $Z(Q_{4}^{\prime})$. As before, to solve $Q_{4}^{\prime}=0$ there are two cases: similarly to the case above, the case where $x_{4}=0$ does not give a ramification point: indeed assuming $x_{4}=0$ we get $0=x_{1}^{2}+0+\phi x_{0}^{2}$, and the $x_{0}$ discriminant is [math] if and only if both are [math]. In the other case we set $x_{4}^{2}=\phi$, and then $x_{1}^{2}=1-x_{2}^{2}$, and we have to find when the $x_{0}$ discriminant of $0=1-x_{2}^{2}+\phi+x_{0}^{2}$ is [math], which happens when

[TABLE]

To find the pullbacks we solve $Q_{3}$: $0=\phi^{2}+\phi+\phi x_{3}^{2}\Rightarrow x_{3}^{2}=-\phi^{2}$. To find $q_{03}^{k^{\prime}}$ we apply the symmetry switching $Q_{0}$ and $Q_{3}$; i.e. we switch between $x_{0},x_{3}$, and $x_{1},x_{2}$. ∎

Proposition 5.4**.**

Taking indices mod 5, The decomposition of $\mathrm{Sym}^{2}H^{0}(K_{{W}})$ into irreducible representations of $({\mathbb{Z}}/2)^{5}\rtimes D_{5}$ (where the $k$th copy of ${\mathbb{Z}}/2$ acts by $x_{k}\mapsto-x_{k}$) is

[TABLE]

Moreover, all these representations are not isomorphic; not even as projective representations.

Proof.

Taking indices modulo 5, $k\in\{1,2\}$ and given a non-trivial vector $v$ in $\mathrm{span}\{x_{j}x_{j+k}\}_{j=1}^{5}$, assume that the coefficient in of $x_{m}x_{m+k}$ in $v$ is non zero, then the sum of $v$ with its image under $x_{m-1}\mapsto-x_{m-1},x_{m+k+2}\mapsto-x_{m+k+2}$ is twice this coefficient times $x_{m}x_{m+k}$, and the orbit of this element under the group spans $\mathrm{span}\{x_{j}x_{j+k}\}_{j=1}^{5}$.

To show that these two irreducible representations (for $k=1,2$) are not isomorphic, we consider the decomposition to $-1$ and $1$ eigen-spaces under the action of $x_{0}\mapsto-x_{0},x_{2}\mapsto-x_{2}$. Indeed we see that these decomposition are

[TABLE]

i.e. they are not isomorphic.

Finally, the group $({\mathbb{Z}}/2)^{5}$ acts trivially on $\mathrm{span}(\{x_{j}^{2}\}_{j=0}^{4})$, so for this subspace we only have to consider the classical representation theory of dihedral groups. ∎

Corollary 5.5**.**

The projectivization of the span of the 3rd and 4th representations in Proposition 5.4 is $I_{2}({{W}})$.

Proof.

Direct verification. ∎

6. Certifying a divisor is not two-canonical

The object of this section is to provide a certification algorithm for a divisor in $\mathrm{Div}^{16}(W)$ to be non two-canonical, with some assumption on the maximal multiplicity it incurs (specifically, we deal with all cases with multiplicity at most 3). By “describing the algorithm” we mean a combination of three things: we review all the approximate computations need to be carried out, give bounds on how accurate they have to be, and finally describe how to perform the computations.

From here on we identify all the fibers of the tangent bundle of the linear space $H^{0}(K_{{W}})^{*}$ with the tangent space at [math], which is itself isomorphic to $H^{0}(K_{{W}})^{*}$.

Throughout this section we don’t work with points in the dual canonical system $|K_{{W}}|^{*}$, but rather with their pullbacks under the projectivization map $H^{0}(K_{{W}})^{*}\smallsetminus\{0\}\to|K_{{W}}|^{*}$, or with representatives of these points; i.e. points along the fiber of the projectivization map.

A final point to keep in mind is that the object of this section is to build an algorithm to solve a multi-linear algebra problem, and that the prevalent linear algebra libraries which work over the complex numbers work with Hermitian dot products; hence the relevant statements below are sometimes made in this language. For the same reason we sometimes use complex conjugation, denoted by $x\mapsto\overline{x}$.

Lemma 6.1**.**

Let $p\in H^{0}(K_{{W}})^{*}$ be a representative of some point $\hat{p}\in W$, with a vanishing coordinate $j$. Then the pullback of the tangent $T_{\hat{p}}W$ to $H^{0}(K_{{W}})^{*}$ is linearly spanned by two vectors:

• •

The vector $\overline{p}$, which spans the fiber of the projectivization over $\hat{p}$,

• •

and the vector whose only non zero coordinate is $j$, which is orthogonal to it (in either Hermitian or non-Hermitian dot product).

Proof.

The claim about the first vector is immediate. As for the second, note that since all the $Q_{k}$ are of the form $\sum a_{i}x_{i}^{2}$ — i.e. diagonalized — their gradients are of the form

[TABLE]

specifically, their $j$ coordinate is trivial when evaluated at $p$. The orthogonality claim is also immediate. ∎

Proposition-Definition 6.2**.**

Let $p,\hat{p},j$ as in Lemma 6.1 above. Let $\{u_{n}\}_{n=0}^{2}$ be a basis for $I_{2}(W)$, and let $f$ be a homogeneous polynomial on $H^{0}(K_{{W}})^{*}$ which attains a double zero at $\hat{p}\in W$; i.e.

[TABLE]

Define the Lagrangian ${\mathcal{L}}(x_{k},\lambda_{n}):=f-\sum_{n}\lambda_{n}u_{n}$, then $f$ has a triple zero on $W$ at $\hat{p}$ if and only if the $j,j$ entry of the partial Hessian (i.e. the Hessian w.r.t. the coordinates $x_{i}$) $\mathrm{Hess}_{x}{\mathcal{L}}$ is [math].

Proof.

We use a very slight modification of the bordered Hessian argument used in constrained optimization: Let $r(t)$ be an analytic function from the unit disc in ${\mathbb{C}}$ to $H^{0}(K_{{W}})^{*}$, so that $r(0)=p$, and $r(t)$ projects to $W$ under the projectivization map. Denote the gradient of $r(t)$ by $v$, then applying the chain rule we have

[TABLE]

where $-^{T}$ denotes the transpose, and $\cdot$ denotes the coordinate-wise product (and not the Hermitian bi-linear form, i.e. there is no conjugation involved). Likewise we get:

[TABLE]

Since $\nabla f=\sum_{n}\lambda_{n}\nabla u_{n}$ at $p$, we now have

[TABLE]

Hence the function $f$ has a triple [math] at $\hat{p}$ if and only if the right hand side of the above equation is [math], where $v$ is a representative in the fiber of the pullback of $T_{\hat{p}}W$ to $H^{0}(K_{W})$, which is orthogonal to the fiber of the projectivization map. By Lemma 6.1, we are done. ∎

6.3****Efficient computation of the $j,j$ coefficient of the Hessian, given $\nabla f$.

In the notations of Proposition 6.2, taking indices modulo 5, we need to compute the value of the $j,j$ entry of $\sum_{n}\lambda_{n}\mathrm{Hess}\,u_{n}|_{p}$:

The $u_{n}$s we choose are simply $Q_{j-1},Q_{j},Q_{j+1}$. We now have to solve the equation

[TABLE]

Recalling that $Q_{k}=x_{k-1}^{2}+\phi x_{k}^{2}+x_{k+1}^{2}$ (for $k=0,\ldots,5$), we see that only $Q_{j-1}$ (resp. $Q_{j+1}$) is contributing to the $j-2$ (resp $j+2$) entry of the gradient. Hence, $\lambda_{0}$ (resp. $\lambda_{2}$) is the $j-2$ (resp. $j+2$) entry of $\nabla f$ divided by $2$ times the $j-2$ (resp $j+2$) entry of $p$.

In a similar manner, we see that $\lambda_{1}$ can be expressed either as the $j-1$ coefficient of $\nabla f$ divided by $2$ times the $j-1$ coefficient of $p$, minus $\phi\lambda_{0}$, or as the $j+1$ coefficient of $\nabla f$ divided by the $2$ times the $j+1$ coefficient of $p$, minus $\phi\lambda_{2}$.

Proposition 6.4**.**

Endow $\mathrm{Sym}^{2}H^{0}(K_{{W}})$ with the complex vector norm coming from taking the monomial basis on the $x_{i}$’s. Let $q\in\mathrm{Sym}^{2}H^{0}(K_{{W}})$ be a quadric which does not vanish identically on $W$. Let $D\in\mathrm{Div}^{16}(W)$ be the intersection divisor of $Z(q)$ and the canonical model of $W$. Assume $p\in{\mathbb{A}}_{5}\smallsetminus\{0\}$ is a representative of some point in $D\subset|K_{{W}}|^{*}\cong{\mathbb{P}}^{4}$, and let $p^{\prime},q^{\prime}$ be such that $||p-p^{\prime}||<\epsilon,\quad||q-q^{\prime}||<\delta$, then the following properties hold:

1. (1)

$||q^{\prime}(p^{\prime})||\leq 15\left((||p||+\epsilon)^{2}(||q||+\delta)-||p||^{2}||q||\right)$. 2. (2)

If $p$ is a double point of $D$, then the norm of the unitary projection of $\nabla q^{\prime}(p^{\prime})$ on the unitary complement of $\mathrm{span}\{\nabla Q_{j}(p)\}_{j}$ is bounded by $25\left((||p||+\epsilon)(||q||+\delta)-||p||\cdot||q||\right)$. 3. (3)

If $p$ is a triple point of $D$, and $j,\lambda_{n}$ are as in 6.3, and let $\lambda_{n}^{\prime}$ be $\gamma$ approximations of the $Q_{j-1},Q_{j},Q_{j+1}$ coefficients of the projection of $q^{\prime}(p^{\prime})$ on $\mathrm{span}\{\nabla Q_{k}(p)\}$, then the norm of element at the diagonal $j,j$ entry of $\mathrm{Hess}(q^{\prime})-\sum_{n=0}^{2}\lambda_{n}\mathrm{Hess}(Q_{j-1+n})$ is bounded by

[TABLE]

Proof.

Denote the coordinates of $q,q^{\prime}$ by $q_{ik},q^{\prime}_{ik}$, and the coordinates of $p,p^{\prime}$ by $p_{i},p^{\prime}_{i}$. As for the claim 1, observe that

[TABLE]

and apply the triangle inequality.

Denote the unitary projection on the unitary complement of $\mathrm{span}\{\nabla Q_{j}(p)\}_{j}$ by $P^{\perp}$. By Lagrange multipliers, $P^{\perp}\nabla q=0$. Moreover, the projection of the $i$th component of $\nabla q^{\prime}(p^{\prime})-\nabla q(p)$ is given by

[TABLE]

As before, we now apply the triangle inequality; summing the results over $i=0,\ldots,4$, this proves claim 2.

Finally, denote by $\{\lambda_{n}^{\prime\prime}\}_{n=0}^{2}$ the $Q_{j-1},Q_{j},Q_{j+1}$ coefficients of $P^{\perp}\sum_{k}q^{\prime}_{ik}p^{\prime}_{k}$.

we have to bound the norm of the $j,j$ entry of the $5\times 5$ matrix

[TABLE]

We will now bound the norm of each of these three terms separately. As for the first term any entry of $\mathrm{Hess}(q^{\prime}-q)$ is of norm $<2\delta$. Similarly, for the second term, we note that all entries of $Q_{m}$ are of norm $<\phi$, so any entry of $\mathrm{Hess}(Q_{m})$ is of norm $<2\phi$. Finally, to bound the third term we note that

[TABLE]

and that $Q_{m}$ are diagonal quadric forms, so up to a factor of $2$ this is also the bound on the Hessian of the sum. Hence, any entry of the difference above is bounded by $2(\delta+3(\phi\gamma+||\nabla q^{\prime}(p^{\prime})-\nabla q(p)||))$, and we have already bound the last term in the proof of claim 2. Hence, by Proposition-Definition 6.2, we have proved claim 3 ∎

Having answered the “what we have to compute” and “how accurate do we have the compute” questions, we now move to the “how do we compute” questions. Specifically we have to show how we intend to compute the following:

• •

Find the quadrics vanishing on a given set of points in $W\subset{\mathbb{P}}^{4}$,

• •

of the quadrics above, which are non-trivial under the quotient by $I_{2}(W)$,

• •

of the above, which are non-trivial under the projection to the direct sum of tangents space of $W$ at given points,

• •

of the above, which quadrics satisfy the condition for triple points.

We address the first three issues in Proposition 6.6, Corollary 6.7, and Corollary 6.9 respectively. With the heavy lifting already done, in all of these cases it is more complicated to build the computational gadgets then the prove they are the correct ones – this is usually immediate. As for the last issue, by Proposition-Definition 6.2, and since we are dealing with Hessians of quadratic forms, it does not involve solving or setting up any linear system; merely plugging a few numbers we already computed in a known formula.

Notation 6.5**.**

Since we denote coordinates of points by subscripts, we need an alternative notation for indices when using several points. Out alternative is $-_{[i]}$. Note that there is no ambiguity in the case of quadric forms as in that case coordinates are double indices, so as indexed quadrics are used throughout in this paper we will not clutter the notation.

Proposition 6.6**.**

Assuming $\{p_{[i]}\}_{i=1}^{n}$ are representatives of points in $W\subset{\mathbb{P}}^{4}$, let $M_{15}$ be the $n\times 15$ matrix whose rows are the degree $2$ monomials in the coordinates of the points. Let $M_{15}=U_{15}D_{15}V_{15}^{\dagger}$ be the singular value decomposition $M_{15}$, then the rows of $V_{15}^{\dagger}$ corresponding to the singular value [math] — considered as quadrics over ${\mathbb{P}}^{4}$ in lexicographic basis — span the quadrics which vanish on all the $p_{[i]}$s.

Proof.

the right kernel of $M_{15}$ is the set of enveloping quadrics of the points, expressed in lexicographic basis. ∎

Corollary 6.7**.**

Let $M$ be the projection of $M_{15}$ on the space orthogonal — per Proposition 5.4 and Corollary 5.5 — to $I_{2}(W)$, and let $M=UDV^{\dagger}$ be the singular value decomposition $M$, then the rows of $V^{\dagger}$ corresponding to the singular value [math] — considered as quadrics over ${\mathbb{P}}^{4}$ in lexicographic basis — span the two canonical quadrics which vanish on all the $p_{[i]}$s, and are orthogonal (in representation theoretic sense) to $I_{2}(W)$.

Proof.

This follows from Corollary 5.5. ∎

Remark 6.8** (effective computation of gradient of a quadric).**

Recall that if $q=\sum_{k,k^{\prime}}q_{kk^{\prime}}x_{k}x_{k^{\prime}}$ is a quadric form as above, then the $k$th coordinate of $\nabla q|_{p}$ is $2q_{kk}p_{k}+\sum_{k^{\prime}\neq k}q_{kk^{\prime}}p_{k}^{\prime}$.

Corollary 6.9**.**

Let $q=\sum_{i=1}^{k_{1}}\lambda_{i}q_{i}$ be a sum of quadrics represented by the kernel columns in Proposition 6.6, such that the double points of $Z(q)\cap W$ are supported on points in ${\mathbb{P}}^{4}$ represented by $\{p_{[j]}\}_{j=1}^{n_{2}}\subset{\mathbb{C}}^{5}$. Let $M_{2}$ be the $2n_{2}\times k_{1}$ block matrix, whose $2\times k_{1}$ blocks are the dot products of the $\nabla q_{i}|_{p_{[j]}}$ — computed as in Remark 6.8 above — with the two vectors from Lemma 6.1, evaluated at $p_{[j]}$, and let $M_{2}=U_{2}D_{2}V_{2}^{\dagger}$ be the singular value decomposition of $M_{2}$, then $M_{2}$ has a [math] singular value, and $\{\lambda_{i}\}$ is in the span of the rows of $V^{\dagger}$ corresponding to the [math] values.

Proof.

This follows from part 2 in Proposition 6.4, and from the computation of the tangent space in Lemma 6.1. ∎

Remark 6.10**.**

The converse of Proposition 6.4 is not necessarily correct even when $D$ is the sum of four theta characteristics in $O_{160}\choose 4$ and for $\epsilon,\delta=0$. Namely, one cannot use the opposite inequalities to deduce that the sum of four theta characteristics in $O_{160}\choose 4$ is $2K_{{W}}$. The argument is simply insufficient if the four theta characteristics share a common point, and the “tests” for single, double, and triple multiplicity all pass.

The usage of the results of Corollary 6.9 as an “input” for 6.3 makes things even worse in this respect: it assumes a one dimensional space of solutions, which is not guaranteed.

Luckily, the combination of Propositions 6.4 and 4.2 show that one can substitute exact computation in a specific case by approximate computation in many cases.

7. Certified numeric proof of the main theorem

7.1****Floating point representation and operations accurate.

We represent floating point numbers using IEEE-754 double accuracy which has a $52$ bits mantissa; i.e. representation accuracy of $2^{-53}\sim 1.11\cdot 10^{-16}$. IEEE-754 guarantees at most half a bit error (i.e. $2^{-53}$ multiplicative error) for each multiplication, addition, or subtraction.

7.2****Bounding the accuracy of SVD.

All the SVD decomposition we use are accurate in the following sense: Let $A$ be a matrix which occurs in our program (described below), for which we compute the SVD (these matrices sizes are at most $12\times 16$ in Corollary 6.7, at most $8\times 16$ in Corollary 6.9, and at most $15\times 48$ in verifying Corollary 5.5). Denote the output of the floating point SVD by $U,D,V$, then all the entries of $UU^{\dagger}-Id,VV^{\dagger}-Id$ are $10^{-14}$ away from 0, and all the entries of $UDV^{\dagger}-A$ are $3\cdot 10^{-14}$ away from [math], up to the accuracy of the multiplication, addition, and subtraction operations described in 7.1. See the function VerifiedZSVD in the accompanying code.

Notation 7.3** (Using the symmetry of the automorphism group).**

One may use the symmetry group of the curve acting on the theta characteristics to save computations. Using the entire group is somewhat tricky from a computational point of view as the decomposition series has three terms, however, using just the semi-direct product of the bi-elliptic involutions and the rotations on the coordinates is reasonably easy: It is an index two subgroup of the automorphism group, and has a two term decomposition series. This is what we do in practice; see the functions group_mul and group_act_on_theta in the accompanying code. We denote this group by $G_{0}$.

Notation 7.4**.**

For natural $J<40$, let $q,j$ be the integer quotient and remainder of $J$ by 8, where $j$ is uniquely expressed as $j=j_{0}+2j_{1}+4j_{2}$ for $j_{0},j_{1},j_{2}\in\{0,1\}$. Denote by $pt_{J}$ the coordinate-wise rotation to the right by $q$ of

[TABLE]

(where ${\mathfrak{i}},a$ are as in Proposition 5.3). Denote the theta characteristic comprised of $pt_{a},pt_{b},pt_{c},pt_{d}$ by P[a, b, c, d]. Once we enumerate theta characteristics in Proposition-Definition 7.5 below, we denote a set of theta characteristics whose numbers are $a,b,c,\ldots$ by [a,b,c,...].

Proposition-Definition 7.5**.**

The following list is an enumeration of the theta characteristics of $O_{160}$:

In the group $Aut(W)$ orbit of $q_{30}^{\bullet}$:

0:P[8, 12, 37, 39] 1:P[9, 13, 33, 35] 2:P[10, 14, 32, 34] 3:P[11, 15, 36, 38] 4:P[5, 7, 16, 20] 5:P[1, 3, 17, 21] 6:P[0, 2, 18, 22] 7:P[4, 6, 19, 23] 8:P[13, 15, 24, 28] 9:P[9, 11, 25, 29] 10:P[8, 10, 26, 30] 11:P[12, 14, 27, 31] 12:P[21, 23, 32, 36] 13:P[17, 19, 33, 37] 14:P[16, 18, 34, 38] 15:P[20, 22, 35, 39]

16:P[0, 4, 29, 31] 17:P[1, 5, 25, 27] 18:P[2, 6, 24, 26] 19:P[3, 7, 28, 30] 20:P[8, 12, 36, 38] 21:P[9, 13, 32, 34] 22:P[10, 14, 33, 35] 23:P[11, 15, 37, 39] 24:P[4, 6, 16, 20] 25:P[0, 2, 17, 21] 26:P[1, 3, 18, 22] 27:P[5, 7, 19, 23] 28:P[12, 14, 24, 28] 29:P[8, 10, 25, 29] 30:P[9, 11, 26, 30] 31:P[13, 15, 27, 31]

32:P[20, 22, 32, 36] 33:P[16, 18, 33, 37] 34:P[17, 19, 34, 38] 35:P[21, 23, 35, 39] 36:P[0, 4, 28, 30] 37:P[1, 5, 24, 26] 38:P[2, 6, 25, 27] 39:P[3, 7, 29, 31] 40:P[8, 12, 33, 35] 41:P[9, 13, 37, 39] 42:P[10, 14, 36, 38] 43:P[11, 15, 32, 34] 44:P[1, 3, 16, 20] 45:P[5, 7, 17, 21] 46:P[4, 6, 18, 22] 47:P[0, 2, 19, 23]

48:P[9, 11, 24, 28] 49:P[13, 15, 25, 29] 50:P[12, 14, 26, 30] 51:P[8, 10, 27, 31] 52:P[17, 19, 32, 36] 53:P[21, 23, 33, 37] 54:P[20, 22, 34, 38] 55:P[16, 18, 35, 39] 56:P[0, 4, 25, 27] 57:P[1, 5, 29, 31] 58:P[2, 6, 28, 30] 59:P[3, 7, 24, 26] 60:P[8, 12, 32, 34] 61:P[9, 13, 36, 38] 62:P[10, 14, 37, 39] 63:P[11, 15, 33, 35]

64:P[0, 2, 16, 20] 65:P[4, 6, 17, 21] 66:P[5, 7, 18, 22] 67:P[1, 3, 19, 23] 68:P[8, 10, 24, 28] 69:P[12, 14, 25, 29] 70:P[13, 15, 26, 30] 71:P[9, 11, 27, 31] 72:P[16, 18, 32, 36] 73:P[20, 22, 33, 37] 74:P[21, 23, 34, 38] 75:P[17, 19, 35, 39] 76:P[0, 4, 24, 26] 77:P[1, 5, 28, 30] 78:P[2, 6, 29, 31] 79:P[3, 7, 25, 27]

, and in the group $Aut(W)$ orbit of $q_{32}^{\bullet}$:

80:P[24, 25, 34, 37] 81:P[28, 29, 32, 39] 82:P[30, 31, 33, 38] 83:P[26, 27, 35, 36] 84:P[2, 5, 32, 33] 85:P[0, 7, 36, 37] 86:P[1, 6, 38, 39] 87:P[3, 4, 34, 35] 88:P[0, 1, 10, 13] 89:P[4, 5, 8, 15] 90:P[6, 7, 9, 14] 91:P[2, 3, 11, 12] 92:P[8, 9, 18, 21] 93:P[12, 13, 16, 23] 94:P[14, 15, 17, 22] 95:P[10, 11, 19, 20]

96:P[16, 17, 26, 29] 97:P[20, 21, 24, 31] 98:P[22, 23, 25, 30] 99:P[18, 19, 27, 28] 100:P[24, 25, 35, 36] 101:P[28, 29, 33, 38] 102:P[30, 31, 32, 39] 103:P[26, 27, 34, 37] 104:P[3, 4, 32, 33] 105:P[1, 6, 36, 37] 106:P[0, 7, 38, 39] 107:P[2, 5, 34, 35] 108:P[0, 1, 11, 12] 109:P[4, 5, 9, 14] 110:P[6, 7, 8, 15] 111:P[2, 3, 10, 13]

112:P[8, 9, 19, 20] 113:P[12, 13, 17, 22] 114:P[14, 15, 16, 23] 115:P[10, 11, 18, 21] 116:P[16, 17, 27, 28] 117:P[20, 21, 25, 30] 118:P[22, 23, 24, 31] 119:P[18, 19, 26, 29] 120:P[24, 25, 33, 38] 121:P[28, 29, 35, 36] 122:P[30, 31, 34, 37] 123:P[26, 27, 32, 39] 124:P[1, 6, 32, 33] 125:P[3, 4, 36, 37] 126:P[2, 5, 38, 39] 127:P[0, 7, 34, 35]

128:P[0, 1, 9, 14] 129:P[4, 5, 11, 12] 130:P[6, 7, 10, 13] 131:P[2, 3, 8, 15] 132:P[8, 9, 17, 22] 133:P[12, 13, 19, 20] 134:P[14, 15, 18, 21] 135:P[10, 11, 16, 23] 136:P[16, 17, 25, 30] 137:P[20, 21, 27, 28] 138:P[22, 23, 26, 29] 139:P[18, 19, 24, 31] 140:P[24, 25, 32, 39] 141:P[28, 29, 34, 37] 142:P[30, 31, 35, 36] 143:P[26, 27, 33, 38]

144:P[0, 7, 32, 33] 145:P[2, 5, 36, 37] 146:P[3, 4, 38, 39] 147:P[1, 6, 34, 35] 148:P[0, 1, 8, 15] 149:P[4, 5, 10, 13] 150:P[6, 7, 11, 12] 151:P[2, 3, 9, 14] 152:P[8, 9, 16, 23] 153:P[12, 13, 18, 21] 154:P[14, 15, 19, 20] 155:P[10, 11, 17, 22] 156:P[16, 17, 24, 31] 157:P[20, 21, 26, 29] 158:P[22, 23, 27, 28] 159:P[18, 19, 25, 30]

Proof.

This is the content of the function fill_odd_theta in the accompanying code. It applies the group action as in 7.3 on the odd theta characteristics we got in Proposition 5.3. ∎

Theorem 7.6**.**

*Then intersections of $O_{160}\choose 2$ with sets of pairs of Steiner system are distinct $G_{0}$ orbits which break as follows:  

 

number of orbits of this “numerical” type 3 3 12

size of orbit 5 5 40

number of pairs in the intersection ${O_{160}\choose 2}\cap\Sigma_{\alpha}$ 48 32 24.

Proof.

We first note that in order to see 18 orbit representatives one merely has to run the accompanying computer program, which also prints the data in the table above. The function build_all_steiner in the program picks a representative $a:=\{\theta_{1},\theta_{2},\theta_{3},\theta_{4}\}\in{O_{160}\choose 4}$ for each $({\mathbb{Z}}/2)^{4}\rtimes{\mathbb{Z}}/5$ orbit (as explained in 7.3), and performs on it the following computation:

1. (1)

First run the function maybe_two_k in the accompanying code: Let $n\leq 16$ be the number distinct points on $W\subset|K_{{W}}|^{*}$ in the formal sum $\sum_{\theta\in a}\theta$. We compute the $n\times 12$ matrix $M$ from Corollary 6.7, and it’s SVD. The computation errors incurred are the one coming from evaluating representatives of the points $q_{ij}^{k}$ (the error incurred here is in the floating point representation of $\sqrt{\phi}$), computing the degree 2 monomials of coordinates of said points which happens in the function fill_pts, decomposing to irreducible representation as in Proposition 5.4 which happens in the function irrep_decomp, and finally performing the SVD in Corollary 6.7. For all the $M$s we encounter, the computed singular values of $M$ lie in $[0,10^{-14}]\cup[10^{-2},10^{3}]$. 2. (2)

Next, if $M$ has a computed singular value in the first segment above, and if the number of multiple points in $\sum_{\theta\in a}\theta$ is some $n_{2}>0$ we run the function maybe_two_k_at_least_double in the accompanying code: Let $M_{2}$ be the $2n_{2}\times k_{1}$ matrix from Corollary 6.9, then we compute it’s SVD. The computational errors we accumulate on top of the of the ones from the previous step come from the getting back from irreducible representations basis to monomial basis in the function anti_decomp, computation of the gradient as in Remark 6.8 in the function q_grad_at_pt, a dot product of two vectors of length $5$ per Lemma 6.1, and the SVD from Corollary 6.9 itself. For all the $M_{2}$s we encounter, the computed singular values of $M_{2}$ lie in $[0,10^{-13}]\cup[10^{-2},10]$. 3. (3)

Finally, if the singular values from 2 are in $[0,10^{-13}]$, and $p$ is a triple point in $\sum_{\theta\in a}\theta$, then we run the function maybe_two_k_at_least_triple: This function performs the verification in Proposition-Definition 6.2, and in practice the computation in in 6.3. The computational errors accumulated on top of the ones we already have come from the operations in 6.3, and from a gradient computation as in Remark 6.8. For all the computations we encounter, the difference corresponding to the one in 3 in Proposition 6.4 is in the set $[0,10^{-14}]\cup[10^{-2},10]$.

Let $\overline{A}$ be the set of $a$’s so that either $M$ has no singular value in the first segment in 1, or it does but $a$ has double points, and $M_{2}$ has no singular value in the second segment in 2, or it does, but the value computed in 3 is in the second segment. Then by Proposition 6.4, and by 7.1 and 7.2, for all $a\in\overline{A}$, $\sum_{\theta\in a}\theta\neq 2K_{{W}}$. The function build_all_steiner then verifies that the conditions of Proposition 4.2 holds for $A:={O_{160}\choose 4}\smallsetminus\overline{A}$. Hence by Proposition 4.2, the output of build_all_steiner is a partition of $O_{160}\choose 2$ into it’s intersection with the sets of pairs of Steiner systems of the curve $W$. ∎

Proposition 7.7**.**

For the Steiner systems in $6$ of the $12$ orbits of Steiner system described in the rightmost column of Theorem 7.6 (i.e. $240$ such systems), the space $\left(\mathrm{span}\{q_{\{\theta,\theta+\alpha\}}\}_{\theta\in\Sigma_{W,\alpha}}\right)^{\perp}$ is 2 dimensional. The projections of this space on each of the 1st, 2nd, and 5th representations in Proposition 5.4 are non-trivial, and the projections on the 3rd and 4th representations is trivial.

Proof.

In the function build_vc_alpha_Ic2_perp in the accompanying code we show — subject as usual to 7.1 and 7.2 — that for all partial Steiner systems, the singular values of the matrix whose rows are complex norm 1 representative of $\{q_{\{\theta,\theta+\alpha\}}\}_{\theta\in\Sigma_{W,\alpha}}$ are in $[0,10^{-14}]\cup[10^{-2},10]$. Moreover, in the function handle_vc_a_plus_ic2_dim_13 we show that for all the $240$ partial systems above, $13$ of the values are in the second segment. Since the dimension cannot be bigger than $13$, it is exactly $13$.

As for the projections on the irreducible representations, it is clear from Corollary 5.5 that the projections on the 3rd and 4th representations are trivial. As for the other representations, we project, and compute (numerically, and subject again to 7.1) the magnitude is in the segment $[0.3,1]$. ∎

Corollary 7.8**.**

The intersection of the $13$ dimensional $V_{W,\alpha}$ is $I_{2}(W)$.

Proof.

The intersection is orthogonal to the span of union of unitary orthogonal spaces to the $V_{W,\alpha}$s, which, applying the symmetry group and the second part of Proposition 7.7, is the direct sum of the 1st, 2nd, and 5th representations from Proposition 5.4 ∎

Proof of the main theorem.

Follows from Theorem 7.6, Corollary 7.8 and Proposition 2.4 ∎

Having proven the main theorem, we now move to some remarks about implementation details:

7.9****Testing the dedicated code.

From a falsifiability point of view, a software test can merely prove that some code is wrong, which happens if and when the test fails. However, verifying full or intermediate results in ways which are easier to compute than, and independent of, the computations use to achieve the result give a heuristic affirmation to the claim that the code is performing its intended function. All the “framework” (i.e. OS, compiler, libraries) pieces behind the code used in the proof of 7.6 and 7.7 are extensively (to say the least) tested. There is one piece which is not: the dedicated code itself. The tests for this dedicated code are described below, and the claim therein is that they really are both independent from what they verify, and verify things which are far easier to verify than compute.

• •

The verification discussed in 7.2.

• •

In the function fill_pts, we verify that the representatives of the points composing the $160$ theta characteristics evaluate to [math] on the $Q_{i}$s, up to 7.1.

• •

In the function fill_hyperplanes we verify that the theta hyperplanes have double zeroes on all their intersection points with the curve, up to 7.1.

• •

Moving from monomial lexicographic ordering to irreducible representations and back — as in Proposition 5.4 — are inverses of one another up to 7.1. (this is verified only in one direction, in the function anti_decomp).

• •

The partial Steiner system we get satisfy Propositions 4.5, 4.7 and Remark 4.8: The numbers of sizes of partial Steiner system of each size are verified in the main program, the sizes of the conjugacy classes under the group action is verified in the function build_orbit_reps, and the way they “break” between different $O_{ij}-O_{i^{\prime},j^{\prime}}$ is verified in the function verify_and_print_pair_structure.

• •

In Theorem 7.6 we show that — in Proposition 4.2 notations — the points in $A$ which do not have quadruple intersection with the curve satisfy the boundaries in Proposition 6.4; whereas in order to prove Proposition 4.2 we only have to test the points which are not in $A$. Moreover, the bound used for the points in $A$ is order of magnitudes less than what is required by Proposition 6.4. This is verified in the respective functions described in the proof of the theorem.

• •

For any $R$ conjugacy class $a$ in A (in the sense of Proposition 4.2),

[TABLE]

This is verified in the main function.

Remark 7.10** (Supplying a “traditional” witness for the computation).**

Verifying the decomposition of $O_{160}$ into Steiner systems is of course easier than finding it. However, we can do even better: In each (partial) Steiner system, we can find pairs of pairs of theta characteristics, such that the 16 points in ${\mathbb{P}}^{4}$ involved are all distinct. If — on a given Steiner system — the graph of pairs of theta characteristics connected by this property has one connected component, a spanning tree of this graph is a witness for the collection of pairs of theta characteristics being a Steiner system, in the sense that each of the 16 tuples of points need only verification in the sense of 1 in Proposition 6.4 with infinite accuracy (which can be done over finite fields and using the Chinese remainder theorem). Below is such a witness to one of the sets of pairs of the partial Steiner systems from Proposition 7.7, which was found in the function print_single_tree:

{[0,9,22,70],[0,9,24,142],[0,9,25,82],[0,9,26,83],[0,9,27,143],[0,9,31,42],[0,9,88,134], [0,9,94,111],[0,9,114,149],[0,9,130,154],[2,20,30,49],[2,24,49,142],[2,40,49,71],[2,49,91,92], [2,49,108,132],[2,49,112,129],[2,49,150,152],[4,7,140,141],[4,20,30,140],[5,6,80,81], [5,20,30,80],[8,20,30,62],[22,48,60,70]}.

By Corollary 7.8 thi witness proves the theorem.