Lines on smooth polarized $K3$-surfaces

TL;DR

This paper establishes precise maximum counts for lines on smooth polarized K3 surfaces in projective spaces, confirming conjectured bounds for specific degrees and dimensions.

Contribution

It provides sharp bounds on the number of lines on smooth polarized K3 surfaces for each degree, confirming previous conjectures in key cases.

Findings

Maximum of 42 lines on sextic K3 surfaces in P^4

Maximum of 36 lines on octic K3 surfaces in P^5

Confirmed conjectured bounds for these cases

Abstract

For each integer $D\ge3$, we give a sharp bound on the number of lines contained in a smooth complex $2D$-polarized $K3$-surface in $\mathbb{P}^{D+1}$. In the two most interesting cases of sextics in $\mathbb{P}^4$ and octics in $\mathbb{P}^5$, the bounds are $42$ and $36$, respectively, as conjectured in an earlier paper.

Figures

Equations

$$T:=\text{operator@font}\text{slNS}(X{)}^{\perp }\subset H_2(X;\mathbb{Z}),$$

$$\sum _{i=0}^5{z}_i=\sum _{i=0}^5{a}_i{z}_i=\sum _{i=0}^5{a}_i^2{z}_i=0,a_i\in \mathbb{C},$$

$$21⩽M_{\mathbb{C}}(D)⩽24,D>20\text{and}19⩽M_{\mathbb{R}}(D)⩽24,D>16$$

Full text

The author was partially supported by the TÜBİTAK grant 116F211

1. Introduction

All algebraic varieties considered in the paper are over ${\mathbb{C}}$.

1.1. The line counting problem

The paper deals with a very classical algebra-geometric problem, viz. counting straight lines in a projective surface. We confine ourselves to the smooth $2D$-polarized $K3$-surfaces $X\subset\mathbb{P}^{D+1}$, $D\geqslant 3$, and obtain sharp upper bounds on the number of lines. Our primary interest are sextics in $\mathbb{P}^{4}$ ($D=3$) and octics in $\mathbb{P}^{5}$ ($D=4$); however, the same approach gives us a complete answer for all higher degrees/dimensions as well.

Recall (see [Saint-Donat]) that a projectively normal smooth $K3$-surface $X\subset\mathbb{P}^{D+1}$ has projective degree $2D$. Given a smooth embedding $\varphi\colon X\hookrightarrow\mathbb{P}^{D+1}$, the polarization is the pull-back $\varphi^{*}\mathcal{O}_{\mathbb{P}^{D+1}}(1)$ regarded as a class $h\in H_{2}(X;\mathbb{Z})$; one has $h^{2}=2D$. Since each line $l\subset X$ is a $(-2)$-curve, it is uniquely determined by its homology class $[l]\in\mathop{\operator@font\text{\sl NS}}\nolimits(X)\subset H_{2}(X;\mathbb{Z})$; to simplify the notation, we identify lines and their classes. In particular, the set of lines is finite; its dual incidence graph $\mathop{\operator@font Fn}\nolimits X$ is called the Fano graph, and the primitive sublattice $\mathcal{F}_{h}(X)\subset H_{2}(X;\mathbb{Z})$ generated over $\mathbb{Q}$ by $h$ and all lines $l\in\mathop{\operator@font Fn}\nolimits X$ is called the Fano configuration of $X$. This polarized lattice, subject to certain restrictions (see LABEL:th.K3), defines an equilinear family of $2D$-polarized $K3$-surfaces $X\subset\mathbb{P}^{D+1}$; on the other hand, the lattice $\mathcal{F}_{h}(X)$ is usually recovered from the graph $\mathop{\operator@font Fn}\nolimits X$ (see LABEL:s.graphs).

The case $D=2$, i.e., that of spatial quartics, is very classical and well known. The quartic $X_{64}$ maximizing the number of lines was constructed by F. Schur [Schur:quartics] as early as in 1882. The problem kept reappearing here and there ever since, but no significant progress had been made until 1943, when B. Segre [Segre] published a paper asserting that 64 is indeed the maximal number of lines in a smooth spatial quartic. Recently, S. Rams and M. Schütt [rams.schuett] discovered a gap in Segre’s argument (but not the statement); they patched the proof and extended it to algebraically closed fields of all characteristics other than $2$ or $3$. Finally, in [DIS], we gave an alternative proof and a number of refinements of the statement, including the complete classification of all quartics carrying more than 52 lines. In particular, Schur’s quartic is the only one with the maximal number 64 of lines.

All extremal (carrying more than 52 lines) quartics found in [DIS] are projectively rigid, as they are the so-called singular $K3$-surfaces. Recall that a $K3$-surface $X$ is singular if its Picard rank is maximal, $\rho(X):=\mathop{\operator@font rk}\nolimits\mathop{\operator@font\text{\sl NS}}\nolimits(X)=20$. (This unfortunate term is not to be confused with singular vs. smooth projective models of surfaces.) Up to isomorphism, an abstract singular $K3$-surface is determined by the oriented isomorphism type of its transcendental lattice

[TABLE]

which is a positive definite even integral lattice of rank $2$ (see LABEL:s.lattice; the orientation is given by the class $[\omega]\in T\otimes{\mathbb{C}}$ of a holomorphic $2$-form on $X$); we use the notation $X(T)$, see § 1.2. As a follow-up to [DIS] (and also motivated by [Shimada:X56]), in [degt:singular.K3] I tried to study smooth projective models of singular $K3$-surfaces $X(T)$ of small discriminant $\det T$. Unexpectedly, it was discovered that Schur’s quartic $X_{64}$ can alternatively be characterized as the only smooth spatial model minimizing this discriminant: one has $X_{64}\cong X([8,4,8])$ (see § 1.2 for the notation) and $\det T\geqslant 55$ for any other smooth quartic $X(T)\subset\mathbb{P}^{3}$.

After quartics, next most popular projective models of $K3$-surfaces are sextics $X\subset\mathbb{P}^{4}$ and octics $X\subset\mathbb{P}^{5}$, and the results of [degt:singular.K3] extend to these two classes: if a singular $K3$-surface $X(T)$ admits a smooth sextic or octic model, then $\det T\geqslant 39$ or $32$, respectively. In view of the alternative characterization of Schur’s quartic $X_{64}$ discovered in [degt:singular.K3], this classification, followed by a study of the models, suggested a conjecture that a smooth sextic $X\subset\mathbb{P}^{4}$ (respectively, octic $X\subset\mathbb{P}^{5}$) may have at most $42$ (respectively, $36$) lines. This conjecture is proved in the present paper (the cases $D=3,4$ in Theorem 1.2), even though the original motivating observation that discriminant minimizing singular $K3$-surfaces maximize the number of lines fails already for degree $10$ surfaces $X\subset\mathbb{P}^{6}$ (see LABEL:th.min.det).

Each sextic in $\mathbb{P}^{4}$ is a regular complete intersection of a quadric and a cubic. I am not aware of any previously known interesting examples of large configurations of lines in such surfaces. The maximal number $42$ of lines is attained at a $1$-parameter family containing the discriminant minimizing surfaces $X(2,1,20])$ and $X([6,3,8])$.

Most octics in $\mathbb{P}^{5}$ are also regular complete intersections: they are the so-called triquadrics, i.e., intersections of three quadrics. The most well-known example is the Kummer family

[TABLE]

whose generic members contain $32$ lines: famous Kummer’s $16_{6}$ configuration (see, e.g., Dolgachev [Dolgachev:book]). There are four other, less symmetric, configurations of $32$ lines, either rigid or realized by $2$-parameter families. However, $32$ is not the maximum: there also are configurations with $33$, $34$, and $36$ lines (see Table 1 on \autopagereftab.main).

The space of octics contains a divisor composed by surfaces that need one cubic defining equation (see [Saint-Donat] and LABEL:s.polarized below). We call these octics special and show that they do stand out in what concerns the line counting problem. Large Fano graphs of special octics (vs. triquadrics) are described by LABEL:th.special.

1.2. Common notation

We use the following notation for particular integral lattices of small rank (see LABEL:s.lattice):

• •

$\mathbf{A}_{p}$, $p\geqslant 1$, $\mathbf{D}_{q}$, $q\geqslant 4$, $\mathbf{E}_{6}$, $\mathbf{E}_{7}$, $\mathbf{E}_{8}$ are the positive definite root lattices generated by the indecomposable root systems of the same name (see [Bourbaki:Lie]);

• •

$[a]:=\mathbb{Z}u$, $u^{2}=a$, is a lattice of rank $1$;

• •

$[a,b,c]:=\mathbb{Z}u+\mathbb{Z}v$, $u^{2}=a$, $u\cdot v=b$, $v^{2}=c$, is a lattice of rank $2$; when it is positive definite, we assume that $0<a\leqslant c$ and $0\leqslant 2b\leqslant a$: then, $u$ is a shortest vector, $v$ is a next shortest one, and the triple $(a,b,c)$ is unique;

• •

$\mathbf{U}:=[0,1,0]$ is the hyperbolic plane.

Besides, $L(n)$, $n\in\mathbb{Z}$, is the lattice obtained by the scaling of a given lattice $L$, i.e., multiplying all values of the form by a fixed integer $n\neq 0$.

To simplify the statements, we let the girth of a forest equal to infinity, so that the inequality $\mathop{\operator@font girth}\nolimits(\Gamma)\geqslant m$ means that $\Gamma$ has no cycles of length less than $m$.

When describing lists of integers, $a\mathbin{..}b$ means the full range $\mathbb{Z}\cap[a,b]$.

We denote by $\{a\bmod d\}$ the arithmetic progression $a+nd,\ldots$, $n\geqslant 0$, and use the shortcut $\{a_{1},\ldots,a_{r}\bmod d\}:=\bigcup_{i}\{a_{i}\bmod d\}$ for finite unions.

1.3. Principal results

Given a field $K\subset{\mathbb{C}}$ and an integer $D\geqslant 2$, let $M_{K}(D)$ be the maximal number of lines defined over $K$ that a smooth $2D$-polarized $K3$-surface $X\subset\mathbb{P}^{D+1}$ defined over $K$ may have.

Most principal results of the paper are collected in Table 1; precise statements are found in the several theorems below. (For the sake of completeness, we also cite some results of [DIS] concerning quartics, i.e., $D=2$.)

In the table, we list:

• •

the degrees $h^{2}=2D$ for which extra information is available (marking with a † the two values that are special in the sense of LABEL:th.min.det),

• •

the bounds $M:=M(D)$ and $\bar{M}:=\bar{M}(D)$ used in Theorem 1.2 (according to which, $M(D)=M_{\mathbb{C}}(D)$ for all values of $D$ in the table), and

• •

the maximal number $M_{\mathbb{R}}:=M_{\mathbb{R}}(D)$ of real lines (see LABEL:th.main.real).

Then, for each value of $D$, we list, line-by-line, all Fano graphs $\Gamma:=\mathop{\operator@font Fn}\nolimits X$ containing more than $\bar{M}(D)$ lines (the notation is explained below), marking with a ‡ those realized by real lines in real surfaces (see LABEL:th.main.real) and indicating

• •

the order $\mathopen{|}\mathop{\operator@font Aut}\nolimits\Gamma\mathclose{|}$ of the full automorphism group of $\Gamma$ and

• •

the transcendental lattices $T:=\mathcal{F}_{h}(\Gamma)^{\perp}=\mathop{\operator@font\text{\sl NS}}\nolimits(X)^{\perp}\in H_{2}(X;\mathbb{Z})$ of generic smooth $2D$-polarized $K3$-surfaces $X$ with $\mathop{\operator@font Fn}\nolimits X\cong\Gamma$ (marking with a ∗ the lattices resulting in pairs of complex conjugate equilinear families).

For the rigid configurations ($\mathop{\operator@font rk}\nolimits T=2$), we list, in addition,

• •

the determinant $\det T$ (underlining the minimal ones, see LABEL:th.min.det),

• •

the numbers $(r,c)$ of, respectively, real and pairs of complex conjugate projective isomorphism classes of surfaces $X$ with $\mathop{\operator@font Fn}\nolimits X\cong\Gamma$, and

• •

the order $\mathopen{|}\mathop{\operator@font Aut}\nolimits X\mathclose{|}$ of the group of projective automorphisms of $X$.

Each of the few non-rigid configurations $\Gamma$ appearing in the table is realized by a single connected equilinear deformation family $\mathcal{M}_{D}(\Gamma)$; we indicate the dimension \dim\bigl{(}\mathcal{M}_{D}(\Gamma)/\!\mathop{\operator@font\text{\sl PGL}}\nolimits(D+2,{\mathbb{C}})\bigr{)}=\mathop{\operator@font rk}\nolimits T-2 and, when known, the minimum of the discriminants $\det T$ of the singular $K3$-surfaces $X(T)\in\mathcal{M}_{D}(\Gamma)$.

If $D=2$, we use the notation for the extremal configurations introduced in [DIS]; otherwise, we refer to the isomorphism classes of the Fano graphs introduced and discussed in more details elsewhere in the paper. In both cases, the subscript is the number of lines in the configuration. For technical reasons, we subdivide Fano graphs into several classes (see LABEL:s.taxonomy) and study them separately, obtaining more refined bounds for each class. Thus, a Fano graph $\Gamma:=\mathop{\operator@font Fn}\nolimits X$ and the configuration $\mathcal{F}_{h}(\Gamma):=\mathcal{F}_{h}(X)$ are called

• •

triangular (the $\Psi_{*}$-series, see LABEL:th.trig), if $\mathop{\operator@font girth}\nolimits(\Gamma)=3$; all extremal quartics also fall into this class,

• •

quadrangular (the $\Theta_{*}$-series, see LABEL:th.quad), if $\mathop{\operator@font girth}\nolimits(\Gamma)=4$,

• •

pentagonal (the $\Phi_{*}$-series, see LABEL:th.a4), if $\mathop{\operator@font girth}\nolimits(\Gamma)=5$,

• •

astral (the $\Delta_{*}$-series, see LABEL:th.astral), if $\mathop{\operator@font girth}\nolimits(\Gamma)\geqslant 6$ and $\Gamma$ has a vertex $v$ of valency $\mathop{\operator@font val}\nolimits v\geqslant 4$.

All other graphs are locally elliptic (the $\Lambda_{*}$-series, see LABEL:s.locally.elliptic), i.e., one has $\mathop{\operator@font val}\nolimits v\leqslant 3$ for each vertex $v\in\Gamma$.

In our notation for particular graphs/configurations, the subscript always stands for the number of vertices/lines. The precise description of all graphs “named” in the paper is available electronically (in the form of GRAPE records) in [degt:Fano.graphs]; in most cases, the implicit reference to [degt:Fano.graphs] is, in fact, the definition of the graph.

1.4. The bounds

Geometrically, apart from the spatial quartics, the two most interesting projective models of $K3$-surfaces are sextics in $\mathbb{P}^{4}$ and octics (especially triquadrics) in $\mathbb{P}^{5}$. However, it turns out that the structure of the Fano graphs simplifies dramatically when the degree $h^{2}=2D$ grows, and one can easily obtain the sharp bounds $M_{\mathbb{C}}(D)$ and $M_{\mathbb{R}}(D)$ for all values of $D$. Below, these bounds are stated for $D$ small and $D\to\infty$; the rather erratic precise values of

[TABLE]

are postponed till LABEL:s.locally.elliptic (see Corollaries LABEL:cor.le.mc and LABEL:cor.le.mr, respectively).

For the next theorem, given a degree $h^{2}=2D$, let $M:=M(D)$ and $\bar{M}:=\bar{M}(D)$ be as in Table 1 or $M(D)=\bar{M}(D)=24$ if $D$ is not found in the table. As in [DIS], in addition to the upper bound $\mathopen{|}\mathop{\operator@font Fn}\nolimits X\mathclose{|}\leqslant M(D)$, we give a complete classification of all large (close to maximal) configurations of lines.

Theorem 1.2** (see LABEL:proof.main).**

Let $X\subset\mathbb{P}^{D+1}$ be a smooth $2D$-polarized $K3$-surface. Then one has $\mathopen{|}\mathop{\operator@font Fn}\nolimits X\mathclose{|}\leqslant M(D)$. More precisely, $\mathopen{|}\mathop{\operator@font Fn}\nolimits X\mathclose{|}\leqslant\bar{M}(D)$ unless $\Gamma:=\mathop{\operator@font Fn}\nolimits X$ is one of the exceptional graphs (configurations) listed in Table 1.

Both bounds are sharp whenever $2\leqslant D\leqslant 20$ or $D\in\{1,2,4,6,7,10\bmod 12\}$;* in particular, one has $M_{\mathbb{C}}(D)=M(D)$ for these values.*