Numerical invariants of surfaces in ${\mathbb P}^4$ lying on small degree hypersurfaces

Abstract

An analogue of the Ellingsrud–Peskine finiteness result is obtained and the Albanese dimension is studied for smooth surfaces in ${\mathbb P}^4$ of non-negative Kodaira dimension that lie on a hypersurface of degree at most 5.

Bogomolov filtration

For the sake of the exposition and lack of a convenient reference for the Bogomolov filtration of a coherent sheaf, we prove Lemma A.2 below. Let X be a smooth complex projective surface. We denote by ${\text {NS}}(X)$ the Néron–Severi group of X. The intersection product defines an integral quadratic form on ${\text {NS}}(X)$. By the Hodge Index Theorem, its real extension to $N(X)={\text {NS}}(X)\otimes _{\mathbb Z}{\mathbb R}$ is of type $(1,\rho -1)$, with $\rho$ the Picard number of X. The positive cone of X is the open cone

$$\begin{aligned} N^+(X)= & {} \{D\in N(X) \mid D^2>0,H\cdot D>0,\\&\text { for some (hence every) ample divisor class }H\text { on }X\}. \end{aligned}$$

It contains the ample cone and is contained in the cone of effective divisors. For ${\mathcal {F}}$ a coherent sheaf on X of rank $r=r_{\mathcal {F}}$, the discriminant of ${\mathcal {F}}$ is the expression

$$\begin{aligned} \varDelta ({\mathcal {F}}) = 2r\,c_2({\mathcal {F}})-(r-1)\,c_1^2({\mathcal {F}}). \end{aligned}$$

Theorem A.1

(Bogomolov, [5]) Let ${\mathcal {F}}$ be a torsion free coherent sheaf on a surface X. If $\varDelta ({\mathcal {F}})<0$, then there exists a maximal non-trivial saturated subsheaf ${\mathcal {F}}'$ such that

• $\varDelta ({\mathcal {F}}')\ge 0$,

• $\dfrac{c_1({\mathcal {F}}')}{r_{{\mathcal {F}}'}}-\dfrac{c_1({\mathcal {F}})}{r_{{\mathcal {F}}}} \in N^+(X)$ and $\bigg ( c_1({\mathcal {F}}')-\dfrac{r_{{\mathcal {F}}'}}{r_{\mathcal {F}}}\,c_1({\mathcal {F}}) \bigg )^{2} \ge -\dfrac{\varDelta ({\mathcal {F}})}{2r_{\mathcal {F}}}$.

In particular, if ${\mathcal {F}}$ is D-semistable³ with respect to an ample divisor D, then $\varDelta ({\mathcal {F}})\ge 0$.

A torsion free sheaf is called Bogomolov unstable if $\varDelta ({\mathcal {F}})<0$ and Bogomolov semistable if $\varDelta ({\mathcal {F}})\ge 0$. The theorem asserts that a torsion free Bogomolov unstable sheaf contains a maximal Bogomolov semistable subsheaf which destabilizes it with respect to every polarization. Such a subsheaf is called a maximal Bogomolov destabilizing subsheaf of the given sheaf.

Lemma A.2

Let ${\mathcal {F}}$ be a locally free sheaf on the surface X. There exists a unique Bogomolov filtration of ${\mathcal {F}}$,

$$\begin{aligned} 0={\mathcal {F}}_0 \subset {\mathcal {F}}_1 \subset \cdots \subset {\mathcal {F}}_m={\mathcal {F}}\end{aligned}$$

such that for each $1\le i\le m$, ${\mathcal {F}}_i/{\mathcal {F}}_{i-1}$ is the maximal Bogomolov destabilizing subsheaf of ${\mathcal {F}}_j/{\mathcal {F}}_{i-1}$ for every $j>i$.

Proof

We argue by induction on the rank $r={\text {rank}}({\mathcal {F}})$. For $r=1$ the statement is obvious, since by definition locally free sheaves of rank 1 are Bogomolov semistable. So we assume $r\ge 2$ and suppose that the theorem holds for all locally free sheaves of inferior rank. Furthermore, we can assume that ${\mathcal {F}}$ is Bogomolov unstable (since otherwise there is nothing to prove).

Let ${\mathcal {F}}_1$ be a maximal Bogomolov destabilizing subsheaf of ${\mathcal {F}}$. By assumption, ${\mathcal {F}}_1\ne {\mathcal {F}}$. Since ${\mathcal {F}}_1$ is saturated, the quotient ${\mathcal {F}}/{\mathcal {F}}_1$ is torsion free, and therefore ${\mathcal {F}}_1$ is reflexive (see [13, Proposition 5.22]), hence locally free, since X is a surface. Now, if the quotient ${\mathcal {F}}/{\mathcal {F}}_1$ is Bogomolov stable, the filtration reduces to $0={\mathcal {F}}_0\subset {\mathcal {F}}_1\subset {\mathcal {F}}_2={\mathcal {F}}$ and we are done. If not, the quotient ${\mathcal {F}}/{\mathcal {F}}_1$ has the rank strictly smaller than r and hence the theorem holds for (the reflexive hull or the double dual of) ${\mathcal {F}}/{\mathcal {F}}_1$. Hence $\left( {\mathcal {F}}/{\mathcal {F}}_1\right) ^{**}$ admits a unique Bogomolov filtration. Lifting this filtration to ${\mathcal {F}}$ gives the desired filtration of ${\mathcal {F}}$. It is enough to describe the procedure for the lifting of the maximal Bogomolov destabilizing subsheaf, call it ${\mathcal {G}}'$, of ${\mathcal {F}}/{\mathcal {F}}_1$ and then apply it inductively for other pieces of the Bogomolov filtration of $\left( {\mathcal {F}}/{\mathcal {F}}_1\right) ^{**}$.

Let ${\mathcal {G}}''$ be the quotient of the inclusion ${\mathcal {G}}' \subset {\mathcal {F}}/{\mathcal {F}}_1$. We have the diagram

where ${\mathcal {F}}_2$ is the kernel of the epimorphism ${\mathcal {F}}\rightarrow {\mathcal {G}}''$. As before, in this short exact sequence ${\mathcal {F}}$ is locally free and ${\mathcal {G}}''$ is torsion free, hence ${\mathcal {F}}_2$ is locally free. Clearly ${\mathcal {F}}_1\subset {\mathcal {F}}_2$ and ${\mathcal {G}}'\simeq {\mathcal {F}}_2/{\mathcal {F}}_1$. We must show that ${\mathcal {F}}_2$ is Bogomolov unstable and that ${\mathcal {F}}_1$ is a maximal Bogomolov destabilizing subsheaf of ${\mathcal {F}}_2$.

Set $r={\text {rank}}({\mathcal {F}})$, $r_j={\text {rank}}({\mathcal {F}}_j)$, and $r_{{\mathcal {G}}'}={\text {rank}}({\mathcal {G}}')$. Since

$$\begin{aligned} \begin{aligned} \frac{c_1({\mathcal {F}}_2)}{r_2}-\frac{c_1({\mathcal {F}})}{r}&= \bigg (\frac{c_1({\mathcal {F}}_1)}{r_1}-\frac{c_1({\mathcal {F}})}{r}\bigg ) -\bigg (\frac{c_1({\mathcal {F}}_1)}{r_1}-\frac{c_1({\mathcal {F}}_2)}{r_2}\bigg )\\&= \bigg (\frac{c_1({\mathcal {F}}_1)}{r_1}-\frac{c_1({\mathcal {F}})}{r}\bigg ) -\frac{r_{{\mathcal {G}}'}}{r_2}\bigg ( \frac{c_1({\mathcal {F}}_1)}{r_1}-\frac{c_1({\mathcal {G}}')}{r_{{\mathcal {G}}'}} \bigg )\\&= \bigg (\frac{c_1({\mathcal {F}}_1)}{r_1}-\frac{c_1({\mathcal {F}})}{r}\bigg ) -\frac{r_{{\mathcal {G}}'}}{r_2}\bigg ( \frac{c_1({\mathcal {F}}_1)}{r_1}-\frac{c_1({\mathcal {F}}/{\mathcal {F}}_1)}{r-r_1} \bigg )\\&\quad +\frac{r_{{\mathcal {G}}'}}{r_2}\bigg ( \frac{c_1({\mathcal {G}}')}{r_{{\mathcal {G}}'}}-\frac{c_1({\mathcal {F}}/{\mathcal {F}}_1)}{r-r_1} \bigg )\\&= \bigg (1-\frac{r_{{\mathcal {G}}'}}{r_2}\frac{r}{r-r_1}\bigg ) \bigg (\frac{c_1({\mathcal {F}}_1)}{r_1}-\frac{c_1({\mathcal {F}})}{r}\bigg ) +\frac{r_{{\mathcal {G}}'}}{r_2}\bigg ( \frac{c_1({\mathcal {G}}')}{r_{{\mathcal {G}}'}}-\frac{c_1({\mathcal {F}}/{\mathcal {F}}_1)}{r-r_1} \bigg )\\&= \frac{r(r-r_2)}{r(r-r_1)} \bigg (\frac{c_1({\mathcal {F}}_1)}{r_1}-\frac{c_1({\mathcal {F}})}{r}\bigg ) +\frac{r_{{\mathcal {G}}'}}{r_2}\bigg ( \frac{c_1({\mathcal {G}}')}{r_{{\mathcal {G}}'}}-\frac{c_1({\mathcal {F}}/{\mathcal {F}}_1)}{r-r_1} \bigg ), \end{aligned} \end{aligned}$$

we see that $c_1({\mathcal {F}}_2)/r_2-c_1({\mathcal {F}})/r\in N^+(X)$. Hence $\varDelta ({\mathcal {F}}_2)<0$, since otherwise ${\mathcal {F}}_2$ would be a Bogomolov destabilizing subsheaf of ${\mathcal {F}}$ and this contradicts the maximality of ${\mathcal {F}}_1$.

We have constructed a Bogomolov unstable subsheaf ${\mathcal {F}}_2$ of ${\mathcal {F}}$ and we claim that ${\mathcal {F}}_1$ is its maximal Bogomolov destabilizing subsheaf. Indeed, if ${\mathcal {F}}_1$ is not a maximal Bogomolov destabilizing subsheaf of ${\mathcal {F}}_2$, then there exists a Bogomolov semistable (locally free) subsheaf ${\mathcal {F}}'$ such that ${\mathcal {F}}_1\subset {\mathcal {F}}'\subset {\mathcal {F}}_2$ and such that $c_1({\mathcal {F}}')/r'-c_1({\mathcal {F}}_2)/r_2\in N^+(X)$. But then, by the previous argument,

$$\begin{aligned} \frac{c_1({\mathcal {F}}')}{r'}-\frac{c_1({\mathcal {F}})}{r} = \bigg (\frac{c_1({\mathcal {F}}')}{r'}-\frac{c_1({\mathcal {F}}_2)}{r_2}\bigg ) +\bigg (\frac{c_1({\mathcal {F}}_2)}{r_2}-\frac{c_1({\mathcal {F}})}{r}\bigg ) \in N^+(X), \end{aligned}$$

contradicting the maximality of ${\mathcal {F}}_1$. $\square$