Counting lines on projective surfaces

TL;DR

This paper establishes a new upper bound on the number of lines on smooth degree-d surfaces in projective space, improving previous bounds and making some classical arguments rigorous, especially for degrees six and higher.

Contribution

It provides the best known bound for the number of lines on such surfaces for degrees six and above, refining and rigorously justifying Segre's classical results.

Findings

New upper bound on lines for degree-d surfaces with d ≥ 3

Improves and rigorizes Segre's classical bounds

Achieves the best known bounds for d ≥ 6

Abstract

We prove a bound on the number of lines on a smooth degree-d surface in three-dimensional projective space for $d \geq 3$. This bound improves a bound due to Segre and renders some of his arguments rigorous. It is the best known bound for $d \geq 6$.

Equations

$$\ell (X_5)\le 127$$

$$\ell (X_d)\le (d-2)(11d-6)$$

$$\ell (X_d)\le 11d^2-30d+18.$$

$$\ell (X_d)\le d(11d-24)$$

$$\mathrm{supp}(\mathcal{F}(X_d))=\{P\in X_d:\text{mbox}thereexistsalineL\text{mbox}suchthati(P,L.X_d)\ge 4\}$$

$${\mathfrak{t}}^{(j)}:=\sum _{0\le i_1,\dots ,i_j\le 3}\frac{\partial f^j}{\partial w_{i_1}\dots \partial w_{i_j}}(w_0,\dots ,w_3)\cdot z_{i_1}\cdot \dots \cdot z_{i_j}.$$

$${\mathfrak{t}}_P^{(j)}(z_0,\dots ,z_3):={\mathfrak{t}}^{(j)}(p_0,\dots ,p_3,z_0,\dots ,z_3).$$

$$\mathcal{P}:=\mathcal{P}(X_d)\subset X_d\times \text{mbox}G(2,4)$$

$$\mathcal{P}:=\{(P,L):P\in X_d\text{mbox}andthelineL\text{mbox}satisfiestheconditioni(P,L.X_d)\ge 3\},$$

$${\pi }_1:\mathcal{P}\to X_d\text{mbox}and{\pi }_2:\mathcal{P}\to \text{mbox}G(2,4).$$

$${\mathfrak{t}}_P^{(2)}{|}_{T_P{X}_d}\equiv 0\text{mbox}foreverypointP\in C.$$

$$X_d\ne \{P\in X_d:\text{mbox}thereexistsalineL\text{mbox}suchthati(P,L.X_d)\ge 4\}.$$

$$Y_d:=(X_d\times {\mathbb{P}}^3)\cap \bigcap _{j=1}^{j=3}\mathrm{supp}({\mathfrak{T}}^{(j)}).$$

$${\Delta }_{X_d}:=\{(P,P)\in {\mathbb{P}}^3\times {\mathbb{P}}^3:P\in X_d\}.$$

$$L_1\subset V({\mathfrak{t}}_P^{(3)})\text{mbox}iffi(P,L_1.X_d)\ge 4.$$

$${\mathfrak{T}}^{(1)}\cdot {\mathfrak{T}}^{(2)}\cdot {\mathfrak{T}}^{(3)}\cdot (X_d\times {\mathbb{P}}^3)$$

$${\mathfrak{t}}_P^{(3)}{|}_{L_k}\text{mbox}doesnotvanishidentically\text{mbox}fork=1,2.$$

$$\text{mbox}either{\mathfrak{t}}_P^{(3)}{|}_{L_k}\equiv 0\text{mbox}orP\text{mbox}istheuniquezeroof{\mathfrak{t}}_P^{(3)}{|}_{L_k}$$

$$\mathfrak{W}:={\mathfrak{T}}^{(1)}\cdot {\mathfrak{T}}^{(2)}\cdot {\mathfrak{T}}^{(3)}\cdot (X_d\times {\mathbb{P}}^3)-6{\Delta }_{X_d}$$

$$\psi (\mathrm{supp}(\mathfrak{W}))=\{P\in X_d:\text{mbox}thereexistsalineL\text{mbox}suchthati(P,L.X_d)\ge 4\}.$$

$$\pi :X_d\to {\mathbb{P}}^1.$$

$$\{P\in L:V({\mathfrak{t}}_P^{(1)})\cap V({\mathfrak{t}}_P^{(2)})\text{mbox}doesnotconsistoftwodistinctlines\}$$

$$i(P,L^{\prime }.X_d)\ge 4$$

$$\mathcal{F}(X_d):={\psi }_{\ast }(\mathfrak{W}\cdot ({\mathbb{P}}^3\times H))$$

$$\psi (\mathrm{supp}(\mathfrak{W}))=\psi (\mathrm{supp}(\mathfrak{W})\cap ({\mathbb{P}}^3\times H)),$$

$$[{\psi }_{\ast }(\mathfrak{W}\cdot ({\mathbb{P}}^3\times H))]$$

$$[{\mathfrak{T}}^{(1)}]\cdot [{\mathfrak{T}}^{(2)}]\cdot [{\mathfrak{T}}^{(3)}]$$

$${\psi }_{\ast }([\mathfrak{W}]\cdot h_2)={\psi }_{\ast }(([{\mathfrak{T}}^{(1)}]\cdot [{\mathfrak{T}}^{(2)}]\cdot [{\mathfrak{T}}^{(3)}]-6[{\Delta }_{{\mathbb{P}}^3}])\cdot {\psi }^{\ast }[X_d]\cdot h_2).$$

$$[{\Delta }_{{\mathbb{P}}^3}]=h_1^3+h_1^2\cdot h_2+h_1.h_2^2+h_2^3,$$

$${\psi }_{\ast }([\mathfrak{W}]\cdot h_2)$$

Full text

Research partially supported by the National Science Centre, Poland, Opus grant no. 2017/25/B/ST1/00853 (S. Rams)

1. Introduction

In this note we establish a new bound on the number of lines that can lie on a smooth projective surface in $\mathbb{P}^{3}$ of given degree.

Recently there has been quite some interest in configurations of lines on surfaces in $\mathbb{P}^{3}({\mathbb{C}})$ (see e.g. [5], [10], [20], [3], [12], [2]). In particular, the picture of the geometry of line configurations on complex projective quartic surfaces is complete (up to [8, Conjecture 4.7]). The claim that the maximal number of lines on a smooth quartic is $64$ can be found in [18], whereas the first correct proof of that fact is given in [14]. The paper [5] contains a complete classification of smooth complex quartic surfaces with many lines. Finally, lines on complex quartics with singular points are considered in [21], [8].

By contrast, the maximal number of lines on smooth hypersurfaces in $\mathbb{P}^{3}({\mathbb{C}})$ of a fixed degree $d\geq 5$ remains unknown (see [18], [3], [12], [6]). In the case of smooth quintic surfaces the proof of the inequality

[TABLE]

can be found in the recent paper [16], whereas (until now) the best bound for smooth complex surfaces of degree $d\geq 6$ has been the inequality

[TABLE]

that was stated by Segre in [18, $\mathsection$ 4].

Segre’s proof of (1) in [18] is based on various properties of so-called lines of the second kind. In particular, he states that every line of the second kind comes up in the flecnodal divisor with multiplicity two (see the next section for details). Unfortunately, several claims made in [18] are false ([13, $\mathsection$ 3]) and the proof of the claim on the multiplicity of lines of the second kind does not fulfill modern standards of rigor (see Remark 2.9.b).

Our original aim while working on this note was to determine whether Segre’s brilliant idea can be proven using modern algebraic geometry (and in particular, whether it is correct). We were able to show that this is the case, and in fact we improved on his bound. Our result addresses the case of characteristic zero as well that of positive characteristic. Under suitable assumptions on the degree, we can use a result of Voloch [22] to conclude that a general point is not flecnodal, even in positive characteristic. We show:

Theorem 1.1**.**

Let $X_{d}\subset\mathbb{P}^{3}({\mathbb{K}})$ be a smooth surface of degree $d\geq 3$ over a field of characteristic [math] or of characteristic $p>d$. Let $\ell(X_{d})$ be the number of lines that the surface $X_{d}$ contains. Then the following inequality holds

[TABLE]

This result provides the lowest known bound on the number of lines lying on a degree-$d$ surface for $d\geq 6$. Still, the question what is the maximal number of lines on smooth projective surfaces of a fixed degree $d\geq 5$ remains open. Note that the Fermat quartic in characteristic $3$ contains $112$ lines – this shows that the assumption $p>d$ cannot be omitted. (see e.g. [15]).

The first bound on the number of lines on a smooth degree-$d$ surface was stated by Clebsch:

[TABLE]

([4, p. 106]), who used ideas coming from Salmon ([4, p. 95], [17]). While a beautiful modern proof of (3) is given in [6, $\mathsection$ 11.2.1], we follow a different path to arrive at (2) (see Remark 2.9.b), so the only fact from [6] we use is the result, due to McCrory and Shifrin [11], that a general point of the surface $X_{d}$ is not flecnodal (Lemma 2.3). In characteristic zero, a bound on the number of lines on complete intersections of codimension two or more can be derived from the orbibundle Miyaoka-Yau-Sakai inequality, but this approach yields no results on the codimension-one case (see [12, Remark 3 on p. 921]).

We do not believe that the particular bound in Theorem 1.1 is sharp. Note, in particular, that it would be in line with the results on quartic surfaces [14, 15] that surfaces might generally be able to carry more lines when the characteristic is positive.

Convention**.**

In this note we work over an algebraically closed field ${\mathbb{K}}$ of characteristic $p$, where either $p=0$ or $p>d$, where $d\geq 3$ is a fixed integer.

2. Lines as multiple components of the flecnodal curve

Let $X_{d}\subset\mathbb{P}^{3}({\mathbb{K}})$ be a smooth degree-$d$ surface, where $d\geq 3$, and let $f$ be a generator of its ideal ${\mathcal{I}}(X_{d})$.

For a line $L\subset\mathbb{P}^{3}$ we put $\operatorname{i}(P,L.X_{d})$ to denote the order of vanishing of the restriction $f|_{L}$ at the point $P$ if $L\nsubseteq X_{d}$. We define $\operatorname{i}(P,L.X_{d}):=\infty$ when $P\in L\subset X_{d}$.

Recall that a line $L\subset X_{d}$ is called a line of the second kind if it meets every plane curve $\Gamma\in|\mathcal{O}_{X_{d}}(1)-L|$ only in inflection points of the latter (see [18, p. 87], [13]). Otherwise, the line $L$ is called a line of the first kind.

In the proof of Theorem 1.1 the following proposition – and in particular its statement about the multiplicities of the lines of the second kind – plays a crucial role (see [18, p. 90] for Segre’s claim in this direction).

Proposition 2.1**.**

Let $X_{d}\subset\mathbb{P}^{3}({\mathbb{K}})$ be a smooth surface of degree $d>2$, where ${\mathbb{K}}$ is an algebraically closed field of characteristic $p$. Let $p=0$ or $p>d$. Then there exists an (effective) Weil divisor ${\mathcal{F}}(X_{d})\in|{\mathcal{O}}_{X_{d}}(11d-24)|$ such that the equality

[TABLE]

holds, and each line $L\subset X_{d}$ of the second kind appears in ${\mathcal{F}}(X_{d})$ with multiplicity at least two.

In the sequel, we call ${\mathcal{F}}(X_{d})$ the *flecnodal divisor of the surface * $X_{d}$.

The proof of Prop. 2.1 will be preceded by several lemmata. First, we introduce the necessary notation. For $j=1,2,3$ we define polynomials ${\mathfrak{t}}^{(j)}\in{\mathbb{K}}[w_{0},\ldots,w_{3},z_{0},\ldots,z_{3}]$ by the formula

[TABLE]

In order to simplify our notation, given a point $P=(p_{0},\ldots,p_{3})\in{\mathbb{K}}^{4}$ (resp. $P=(p_{0}:\ldots:p_{3})\in\mathbb{P}^{3}$) we put

[TABLE]

Observe that the zero set of the polynomial ${\mathfrak{t}}_{P}^{(1)}$ (resp. ${\mathfrak{t}}_{P}^{(2)}$) is the projective tangent space ${T}_{P}X_{d}$ (resp. the Hessian quadric $\mbox{V}_{P}=\mbox{V}_{P}X_{d}$).

We consider the variety

[TABLE]

defined as

[TABLE]

where $\mbox{G}(2,4)$ is the Grassmanian of lines in $\mathbb{P}^{3}({\mathbb{K}})$. The variety ${\mathcal{P}}$ is endowed with the projections

[TABLE]

Lemma 2.2**.**

(a)* If $P\in X_{d}$, then $\#\pi_{1}^{-1}(P)\in\{1,2,\infty\}$.*

(b)* The set $\{P\in X_{d}\,:\,\#\pi_{1}^{-1}(P)=\infty\}$ is finite.*

Proof.

(a) We fix a point $P\in X_{d}$. Obviously, either ${\mathfrak{t}}_{P}^{(2)}|_{{T}_{P}X_{d}}\equiv 0$ or ${\mathfrak{t}}_{P}^{(2)}|_{{T}_{P}X_{d}}$ vanishes along exactly one or two lines.

Let $L\subset{T}_{P}X_{d}$ be a line. By direct computation, ${\mathfrak{t}}_{P}^{(2)}$ vanishes along $L$ if and only if $\operatorname{i}(P,L.X_{d})\geq 3$. Thus a finite fiber $\pi_{1}^{-1}(P)$ consists of at most two points, which yields (a).

(b) Suppose that there exists a curve $C\subset X_{d}$ such that

[TABLE]

Then the Gauss map is constant on the curve $C$, because its differential vanishes for all $P\in\mbox{reg}(C)$. Thus the curve $C$ is contained in a fiber of the Gauss map, which is impossible by Zak’s result [23, Thm. 2.3]. ∎

In particular, we have shown that the variety ${\mathcal{P}}$ is two-dimensional.

We will need to know that not all points on a smooth surface can be flecnodal. In characteristic $p=0$ this result is due to McCrory and Shifrin (see [11, Lemma 2.10], [6, Prop. 11.8]), whereas for characteristic $p>d$ the lemma below was shown by Voloch (see [22, Theorem 1 and Prop. 1]):

Lemma 2.3**.**

Let $X_{d}\subset\mathbb{P}^{3}({\mathbb{K}})$ be a smooth surface of degree $d>2$, where ${\mathbb{K}}$ is an algebraically closed field of characteristic $p$, and let $p=0$ or $p>d$. Then

[TABLE]

In the sequel, the divisor defined by ${\mathfrak{t}}^{(j)}$ in $\mathbb{P}^{3}\times\mathbb{P}^{3}$ is denoted by ${\mathfrak{T}}^{(j)}$ and we put

[TABLE]

Moreover, $\psi\,:\,\mathbb{P}^{3}\times\mathbb{P}^{3}\rightarrow\mathbb{P}^{3}$ stands for the projection onto the first factor and we define

[TABLE]

Lemma 2.4**.**

The variety $Y_{d}$ is two-dimensional.

Proof.

By Lemma 2.2.b, only finitely many fibers of the projection $\psi|_{Y_{d}}$ are two-dimensional.

Fix a point $P\in X_{d}$ such that the fiber $\pi_{1}^{-1}(P)$ is finite, i.e., ${\mathfrak{t}}_{P}^{(2)}$ vanishes on ${T}_{P}X_{d}$ only along two (not necessarily distinct) lines $L_{1},L_{2}$. By direct computation

[TABLE]

Thus, by Lemma 2.3, for generic choice of $P\in X_{d}$ the fiber of $(\psi|_{Y_{d}})^{-1}(P)$ is finite. ∎

Lemma 2.4 implies that the proper intersection $2$-cycle

[TABLE]

is well-defined (see e.g. [19, Chap. V.C.2]). One of its components is the variety $\Delta_{X_{d}}$.

Lemma 2.5**.**

The diagonal $\Delta_{X_{d}}$ comes up in the intersection cycle (6) with multiplicity $6$.

Proof.

We are to show that, for generic choice of the point $P\in X_{d}$, the intersection multiplicity of the curves $V({\mathfrak{t}}_{P}^{(j)}|_{{T}_{P}X_{d}})$, where $j=2,3$, in $P$ equals $6$. By Lemma 2.2.(b) we can assume that $V({\mathfrak{t}}_{P}^{(2)}|_{{T}_{P}X_{d}})$ consists of two lines $L_{1},L_{2}$. Moreover, Lemma 2.3 and (5) allow us to require that

[TABLE]

Then, by direct computation, the restriction ${\mathfrak{t}}_{P}^{(3)}|_{L_{k}}$ has a triple root in $P$ for $k=1,2$ and the proof is complete. ∎

Given a principal line $L_{k}\subset T_{P}X_{d}$, the above proof shows that

[TABLE]

In particular, Lemma 2.5, all components appear in the cycle

[TABLE]

with non-negative coefficients. Moreover, by definition, the set $\psi(\operatorname{supp}({\mathfrak{W}}))$ consists of the points $P$ such that ${\mathfrak{t}}_{P}^{(1)}$, $\ldots$, ${\mathfrak{t}}_{P}^{(3)}$ vanish simultaneously along a line. From (5) and (7), we obtain the equality

[TABLE]

Let $L\subset X_{d}$ be a line. Recall that the linear system $|{\mathcal{O}}_{X_{d}}(1)-L|$ endows the surface in question with a fibration

[TABLE]

Let us follow [18] and put $\Gamma_{P}$ to denote its fiber that is contained in the tangent space ${T}_{P}X_{d}$ for a point $P\in L$. One can easily check that if the Hessian quadric $\mbox{V}_{P}$ does not contain the tangent space in question, then the line residual to $L$ in the (scheme-theoretic) intersection ${T}_{P}X_{d}\cap V_{P}$ is tangent to the curve $\Gamma_{P}$ in the point $P$.

For the proof of Prop. 2.1 we will need the following observation.

Lemma 2.6**.**

Let $L\subset X_{d}$ be a line. Then the set

[TABLE]

is finite.

Proof.

Observe that our assumption on the base field ${\mathbb{K}}$ combined with [9, Prop. IV.2.5] implies that the map $\pi|_{L}\,:\,L\rightarrow\mathbb{P}^{1}$ (i.e., the restriction of the fibration $\pi$ to the line $L$) is separable.

Let $P\in L$ be a point. By Lemma 2.2, we can assume that the Hessian quadric $V_{P}$ does not contain the tangent space ${T}_{P}X_{d}$. If the Hessian quadric and the tangent space meet along the line $L$ with multiplicity two, then $L$ is tangent to the curve $\Gamma_{P}$ in the point $P$. Thus $P$ is the ramification point of the degree-$(d-1)$ map $\pi|_{L}$ (one can repeat verbatim the proof of [16, Lemmata 3.1, 3.2]). But $\pi|_{L}$ is generically etale, so it has only finitely many ramification points, and the claim follows. ∎

One has the following property of lines of the second kind.

Lemma 2.7**.**

Let $L\subset X_{d}$ be a line and let $P\in L$ be a point such that the Hessian quadric $\mbox{V}_{P}$ does not contain the tangent space ${T}_{P}X_{d}$. If the line $L$ is of the second kind, then the form ${\mathfrak{t}}_{P}^{(3)}$ vanishes along the (set-theoretic) intersection ${T}_{P}X_{d}\cap V_{P}$.

Proof.

We can assume that the plane and the quadric meet along two distinct lines, one of which is not contained in $X_{d}$ (otherwise the claim is obvious, because ${\mathfrak{t}}_{P}^{(3)}$ vanishes along $L\subset X_{d}$).

Let $L^{\prime}$ be the line residual to $L$ in ${T}_{P}X_{d}\cap V_{P}$. As we already explained, the line $L^{\prime}$ is tangent to the fiber $\Gamma_{P}$ of $\pi$ in the point $P$. Since $L$ is of the second kind, $P$ is an inflection point of the curve $\Gamma_{P}$, so $L^{\prime}$ meets $\Gamma_{P}$ with multiplicity at least $3$ in the point $P$. But $L\subset X_{d}$ also meets $L^{\prime}$ in the point $P$, so we have

[TABLE]

The claim follows directly from (5). ∎

Now we are in position to give a proof of Prop. 2.1. In the proof below we maintain the notation of this section. In particular the cycle ${\mathfrak{W}}$ is given by (8), and $\psi$ denotes the projection $\mathbb{P}^{3}\times\mathbb{P}^{3}\rightarrow\mathbb{P}^{3}$ onto the first factor.

Proof of Prop. 2.1.

Let $H\subset\mathbb{P}^{3}$ be a generic hyperplane. We claim that the effective divisor

[TABLE]

has the required properties.

Indeed, one can easily see that

[TABLE]

so (9) implies that the support of ${\mathcal{F}}(X_{d})$ satisfies the claim of the proposition.

In order to show that ${\mathcal{F}}(X_{d})\in|{\mathcal{O}}_{X_{d}}(11d-24)|$ we compute the class

[TABLE]

in the Chow ring $A^{*}(\mathbb{P}^{3})$. We put $h_{1}:=[H\times\mathbb{P}^{3}]$ and $h_{2}:=[\mathbb{P}^{3}\times H]$. At first we compute in $\mbox{A}^{*}(\mathbb{P}^{3}\times\mathbb{P}^{3})$:

[TABLE]

Since $[\Delta_{\mathbb{P}^{3}}].\psi^{*}([X_{d}])=[\Delta_{X_{d}}]$ we have

[TABLE]

Recall that, by [7, Ex. 8.4.2], the class $[\Delta_{\mathbb{P}^{3}}]$ of the diagonal in $\mathbb{P}^{3}\times\mathbb{P}^{3}$ can be expressed as

[TABLE]

whereas $h_{1}=\psi^{*}[H]$. Therefore, from (10), we obtain that

[TABLE]

and the proof of that part of the proposition is complete.

Finally, let $L\subset X_{d}$ be a line of the second kind and let $P\in L$ be a point such that ${T}_{P}X_{d}$ and the Hessian quadric $V_{P}$ meet along two distinct lines. As in Lemma 2.7 we put $L^{\prime}$ to denote the line residual to $L$ in ${T}_{P}X_{d}\cap V_{P}$. We can assume that the hyperplane $H$ meets $L$ (resp. $L^{\prime}$) in the point $Q\neq P$ (resp. $Q^{\prime}\neq P$). Obviously we have $(P,Q)\in{\operatorname{supp}({\mathfrak{W}})}\cap(\mathbb{P}^{3}\times H)$. Moreover, by Lemma 2.7, the point $(P,Q^{\prime})$ also belongs to the set ${\operatorname{supp}({\mathfrak{W}})}\cap(\mathbb{P}^{3}\times H)$. Since $Q\neq Q^{\prime}$, Lemma 2.6 implies that the restriction of the projection $\psi$,

[TABLE]

is of degree at least two, so the claim on the multiplicity follows from the definition of the map $\psi_{*}$. ∎

Example 2.8**.**

An elementary computation shows that the Schur quartic

[TABLE]

contains exactly $64$ lines: $48$ lines of the first kind and $16$ lines of the second kind. Since the flecnodal divisor of a quartic surface has degree $80$, each line of the second kind must come in the flecnodal divisor precisely with multiplicity two. Thus, the lower bound of Prop. 2.1 is sharp.

Remark 2.9**.**

(a) The idea of studying lines on a surface via points of fourfold contact goes back to work of Salmon and Clebsch on cubic surfaces (see [10] and the bibliography therein). In particular, an equation of the flecnodal divisor is obtained in [4] via projection of the intersection of the varieties ${\mathfrak{T}}^{(j)}$. A beautiful exposition of a modern treatment of this approach can be found in [6, $\mathsection$ 11.2.1]. Still, for the proof of Prop. 2.1, we find it more convenient to avoid the use of bundles of relative principal parts. In this way we can control the behaviour of the flecnodal divisor along a line of the second kind.

(b) As we already explained, the claim on multiplicities of lines of the second kind in the flecnodal divisor was stated in [18, p. 90]. Segre (see [18, (7) on p. 88]) justified it by giving an explicit formula for an analytic function (defined on an open neighbourhood of a point $P$ on a line $L\subset X_{d}$) that vanishes along the set $\mbox{supp}({\mathcal{F}}(X_{d}))$ and showing that the function in question has multiplicity at least two along the line $L$ provided the latter is of the second kind. Unfortunately, this argument does not explain why the function in question is a local equation of the flecnodal divisor (although it explains why its set of zeroes contains the support $\mbox{supp}({\mathcal{F}}(X_{d}))$), i.e., it does not explain why its order of vanishing along the line $L$ yields any information on the multiplicity with which $L$ comes up in the divisor ${\mathcal{F}}(X_{d})$.

3. Bound on the number of lines

We recall the following fact that we need for the proof of Thm. 1.1.

Claim 3.1**.**

([18, p. 88])* Assume that the characteristic $p$ of the ground field is either zero or bigger than $d$. A line $L\subset X_{d}$ of the first kind is met by at most $(8d-14)$ other lines lying on the surface $X_{d}$.*

Given a (Weil) divisor $Z=\sum_{j}\alpha_{j}C_{j}$ on $X_{d}$ and a plane $\Pi\subset\mathbb{P}^{3}({\mathbb{K}})$ we introduce the following notation:

[TABLE]

For the proof of Thm. 1.1 we need the following observation.

Observation 3.2**.**

Let $L_{1}$, $\ldots$, $L_{k}\subset\mathbb{P}^{3}$ be coplanar lines with $k\leq d$ and let $\Pi=\mbox{span}(L_{1},L_{2})$ be the plane they span. Moreover, assume that $\mbox{supp}(Z)$ contains none of the lines $L_{1}$, $\ldots$, $L_{k}$. Then, the following inequality holds

[TABLE]

Proof.

Obviously, we have the equalities $\deg(Z_{\Pi})=Z_{\Pi}\cdot L_{j}$ for $j=1,\ldots,k$, and the inequality $\deg(Z^{\Pi})\geq Z^{\Pi}.(L_{1}+\ldots+L_{k})$. Therefore, we obtain

[TABLE]

Since $Z=Z^{\Pi}+Z_{\Pi}$, the claim follows. ∎

Now we put:

[TABLE]

Moreover, we assume $h\leq d$ to be a positive integer.

Our aim in this section is to prove:

Main Lemma 3.3**.**

[TABLE]

The Main Lemma then immediately implies Theorem 1.1 upon using the equality $\deg{\mathcal{Z}}=\deg{\mathcal{F}}(X_{d})-\ell(X_{d})$.

Definition 3.1**.**

1. a)

We put $\ell_{1}(X_{d})$ (resp. $\ell_{2}(X_{d})$) to denote the number of lines of that come up with multiplicity one (resp. higher than one) in ${\mathcal{F}}(X_{d})$. 2. b)

We call a line of multiplicity one in ${\mathcal{F}}(X_{d})$ reduced. 3. c)

We call a plane $\Pi$ $k$**-spanned ** if it contains $k$ reduced lines (so each $3$-spanned plane is $2$-spanned etc.).

Obviously, we have

[TABLE]

As a consequence of Obs. 3.2 we obtain the following bound on $\deg(\mathcal{Z})$.

Observation 3.4**.**

Let $L_{1}$, $\ldots$, $L_{k}\subset\mathbb{P}^{3}$ be coplanar reduced lines with $k\leq d$ and let $\Pi=\mbox{span}(L_{1},L_{2})$ be the plane they span. Then, the following inequality holds

[TABLE]

Proof.

The lines $L_{1}$, $\ldots$, $L_{k}$ are reduced, so they are no components of the support of ${\mathcal{Z}}$ and we can apply Obs.  3.2 to obtain

[TABLE]

Recall that $L^{2}=-(d-2)$ for each line $L\subset X_{d}$. Thus for $q=1,\ldots,k$ we have

[TABLE]

By Prop.  2.1 the lines $L_{1}$, $\ldots$, $L_{k}$ are of the first kind. Therefore, by Claim 3.1, for $q=1,\ldots,k$, we have

[TABLE]

and the claim (12) follows. ∎

Observation 3.5**.**

Let $L_{1}$, $\ldots$, $L_{k}\subset\mathbb{P}^{3}$ be coplanar reduced lines with $2\leq k\leq d$ and let $\Pi=\mbox{span}(L_{1},L_{2})$ be the plane they span. Moreover, let

[TABLE]

Then the following implication holds:

[TABLE]

Proof.

Insert the assumption (13) into (12). ∎

Lemma 3.6**.**

Let $L_{1}$ be a reduced line.

1. a)

If there exist three $2$-spanned planes $\Pi_{1}$, $\Pi_{2}$, $\Pi_{3}$ that meet along $L_{1}$, then

[TABLE] 2. b)

If there exist two $3$-spanned planes $\Pi_{1}$, $\Pi_{2}$, that meet along $L_{1}$, then

[TABLE]

Proof.

a) If $\deg({\mathcal{Z}}_{\Pi_{j}})\leq\frac{(4\cdot 2-6)(d-3)}{2-1}$ for one of the planes $\Pi_{1}$, $\Pi_{2}$, $\Pi_{3}$, then $\deg({\mathcal{Z}})\geq 6(d-3)$ by Obs. 3.5 and the proof is complete.

Assume that

[TABLE]

then

[TABLE]

and the proof is complete.

b) The claim follows as in part a). ∎

In the sequel we will also need the following bound on $\deg({\mathcal{Z}})$.

Lemma 3.7**.**

Let $L_{1}$ be a reduced line. Assume that $L_{1}$ is met by at most $q$ other reduced lines. Then

[TABLE]

Proof.

Obviously, we have $({\mathcal{F}}(X_{d})-L_{1})\cdot L_{1}=12d-26$, so this gives the following bound on the number of other lines on $X_{d}$ that meet $L_{1}$:

[TABLE]

because each non-reduced line comes with multiplicity at least two in the flecnodal divisor. We have

[TABLE]

Thus

[TABLE]

and the claim follows. ∎

Lemma 3.8**.**

Let $L_{1}$ be a reduced line. Assume that $L_{1}$ is met by at most $q_{1}$ other reduced lines and $q_{2}$ non-reduced lines. Then

[TABLE]

Proof.

Since we have

[TABLE]

we get

[TABLE]

∎

We can now prove the Main Lemma 3.3.

Proof of Lemma 3.3..

If $X_{d}$ contains at least $6(d-3)$ lines that are not reduced, then the claim follows from the inequality (11). Therefore for the rest of the proof we assume that the following inequality holds:

[TABLE]

Step 1: Assume there are no reduced lines on $X_{d}$. Then

[TABLE]

Step 2: Suppose there exists a reduced line that is met by at most $10$ reduced lines. Then Lemma 3.7 yields Lemma 3.3. Thus we can assume the following for the remainder of the proof.

• Assumption A.* There exists a reduced line on $X_{d}$ and each reduced line on $X_{d}$ is met by at least 11 other reduced lines.*

Step 3: If there exists a reduced line on $X_{d}$ that is contained either in three $2$-spanned planes or in two $3$-spanned planes, then Lemma 3.6 yields the claim. Thus we can assume that each reduced line $L_{1}\subset X_{d}$ forms part of one of the two configurations:

1. (1)

$L_{1}$ is contained in exactly one plane that is spanned by reduced lines and, by Assumption A, the plane in question is $k$-spanned with $k\geq 12$, 2. (2)

$L_{1}$ is contained in exactly two planes that are spanned by reduced lines and one of them fails to be $3$-spanned (it is only $2$-spanned).

Step 4: Suppose there exists a reduced line $L_{1}$ that is contained in exactly one $k$-spanned plane $\Pi$, i.e. all reduced lines that meet $L_{1}$ are contained in the plane $\Pi$. Let $L_{2}$, $\ldots$, $L_{k}$, with $k\geq 12$, be the reduced lines that meet the line $L_{1}$.

Each of the lines $L_{2}$, $\ldots$, $L_{k}$ is contained in at most two $2$-spanned planes, (and if it is contained in a $2$-spanned plane $\neq\Pi$, then the other plane is $2$-spanned but not $3$-spanned) by Lemma 3.6. Thus each of the lines $L_{2}$, $\ldots$, $L_{k}$ is met by at most one reduced line that is not contained in $\Pi$.

At most $(d-k)$ non-reduced lines are contained in $\Pi$ and we have at most $6(d-3)-1$ non-reduced lines on $X_{d}$. Our surface is smooth, so a line that is not contained in $\Pi$ meets at most one of the lines $L_{1}$, $\ldots$, $L_{k}$ (otherwise it would be in $\Pi$, or it would meet two lines $L_{j_{1}},L_{j_{2}}$, where $j_{1}\neq j_{2}$, in the same point and the latter would be a singularity of $X_{d}$ because the tangent space of $X_{d}$ would be too large).

Thus one of the lines $L_{1}$, $\ldots$, $L_{k}$ is met by at most

[TABLE]

non-reduced lines. Lemma 3.8 yields (with $q_{1}=k-1$ and $q_{2}<(d-k)+\frac{1}{2}(d-3)$) the inequality

[TABLE]

which implies Lemma 3.3. Thus we can assume the following.

• Assumption B.* Each reduced line is contained in exactly two $2$-spanned planes, one of which is not $3$-spanned (but only $2$-spanned).*

Step 5: Let $L_{1}$ be a reduced line and let it be met by $k$ other reduced lines. By Assumption B the line $L_{1}$ is contained in two $2$-spanned planes: $\Pi_{1}$ and $\Pi_{2}$. We can assume $\Pi_{2}=\mbox{span}(L_{1},L_{k+1})$ to be $2$-spanned, but not $3$-spanned. Then $\Pi_{1}=\mbox{span}(L_{1},\ldots,L_{k})$ is $k$-spanned but not $(k+1)$-spanned, where $L_{1}$, $\ldots$, $L_{k+1}$ are assumed to be reduced lines. By Assumption A we have $k\geq 11$.

At most $(d-k)$ non-reduced lines are contained in $\Pi_{1}$ and we have at most $6(d-3)-1$ non-reduced lines on $X_{d}$. Again, one of the lines $L_{1}$, $\ldots$, $L_{k}$ is met by at most

[TABLE]

non-reduced lines. Moreover it is met by exactly $k$ reduced lines. As in Step 4 we obtain

[TABLE]

and the proof is complete. ∎

Finally, we can give the proof of Theorem 1.1.

Proof of Theorem 1.1..

The claim follows immediately from the Main Lemma 3.3 and the equality $\deg{\mathcal{Z}}=\deg{\mathcal{F}}(X_{d})-\ell(X_{d})$. ∎

Example 3.9**.**

The Fermat surface

[TABLE]

contains $3d^{2}$ lines. For $d\neq 4,6,8,12,20$ this is up to now the best example of a surface with many lines (see e.g. [3]).

Acknowledgement. We would like to acknowledge the big impact of Wolf Barth on our research. He introduced us to the circle of ideas surrounding rational curves on surfaces while we were postdoctoral students in Erlangen. This note would have never been written without his generous support and the inspiring ideas [1] that he communicated to us at that time.