A lower bound for $\chi ({\mathcal {O}}_S)$

Abstract

Let $(S,{\mathcal {L}})$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle ${\mathcal {L}}$ of degree $d > 25$. In this paper we prove that $\chi (\mathcal O_S)\ge -\frac{1}{8}d(d-6)$. The bound is sharp, and $\chi ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$ if and only if d is even, the linear system $|H^0(S,{\mathcal {L}})|$ embeds S in a smooth rational normal scroll $T\subset {\mathbb {P}}^5$ of dimension 3, and here, as a divisor, S is linearly equivalent to $\frac{d}{2}Q$, where Q is a quadric on T. Moreover, this is equivalent to the fact that a general hyperplane section $H\in |H^0(S,{\mathcal {L}})|$ of S is the projection of a curve C contained in the Veronese surface $V\subseteq {\mathbb {P}}^5$, from a point $x\in V\backslash C$.

1 Introduction

In [6] D. Franco and the author prove a sharp lower bound for the self-intersection $K^2_S$ of the canonical bundle of a smooth, projective, complex surface S, polarized by a very ample line bundle ${\mathcal {L}}$, in terms of its degree $d={\text {deg}}\,{\mathcal {L}}$, assuming $d>35$. Refining the line of the proof in [6], in the present paper we deduce a similar result for the Euler characteristic $\chi ({\mathcal {O}}_S)$ of S [1, p. 2], in the range $d>25$. More precisely, we prove the following:

Theorem 1.1

Let $(S,{\mathcal {L}})$ be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle ${\mathcal {L}}$ of degree $d > 25$. Then:

$$\begin{aligned} \chi ({\mathcal {O}}_S)\ge -\frac{1}{8}d(d-6). \end{aligned}$$

The bound is sharp, and the following properties are equivalent.

(i) $\chi ({\mathcal {O}}_S)= -\frac{1}{8}d(d-6)$;

(ii) $h^0(S,{\mathcal {L}})=6$, and the linear system $|H^0(S,\mathcal L)|$ embeds S in ${\mathbb {P}}^5$ as a scroll with sectional genus $g=\frac{1}{8}d(d-6)+1$;

(iii) $h^0(S,{\mathcal {L}})=6$, d is even, and the linear system $|H^0(S,{\mathcal {L}})|$ embeds S in a smooth rational normal scroll $T\subset {\mathbb {P}}^5$ of dimension 3, and here S is linearly equivalent to $\frac{d}{2}(H_T-W_T)$, where $H_T$ is the hyperplane class of T, and $W_T$ the ruling (i.e. S is linearly equivalent to an integer multiple of a smooth quadric $Q\subset T$).

By Enriques’ classification, one knows that if S is unruled or rational, then $\chi ({\mathcal {O}}_S)\ge 0$. Hence, Theorem 1.1 essentially concerns irrational ruled surfaces.

In the range $d>35$, the family of extremal surfaces for $\chi ({\mathcal {O}}_S)$ is exactly the same for $K^2_S$. We point out there is a relationship between this family and the Veronese surface. In fact one has the following:

Corollary 1.2

Let $S\subseteq {\mathbb {P}}^r$ be a nondegenerate, smooth, irreducible, projective, complex surface, of degree $d > 25$. Let $L\subseteq {\mathbb {P}}^r$ be a general hyperplane. Then $\chi ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$ if and only if $r=5$, and there is a curve C in the Veronese surface $V\subseteq \mathbb P^5$ and a point $x\in V\backslash C$ such that a general hyperplane section $S\cap L$ of S is the projection $p_x(C)\subseteq L$ of C in $L\cong {\mathbb {P}}^4$, from the point x.

In particular, $S\cap L$ is not linearly normal, even if S is.

2 Proof of Theorem 1.1

Remark 2.1

(i) We say that $S\subset {\mathbb {P}}^r$ is a scroll if S is a ${\mathbb {P}}^1$-bundle over a smooth curve, and the restriction of ${\mathcal {O}}_S(1)$ to a fibre is $\mathcal O_{{\mathbb {P}}^1}(1)$. In particular, S is a geometrically ruled surface, and therefore $\chi ({\mathcal {O}}_S)= \frac{1}{8}K^2_S$ [1, Proposition III.21].

(ii) By Enriques’ classification [1, Theorem X.4 and Proposition III.21], one knows that if S is unruled or rational, then $\chi ({\mathcal {O}}_S)\ge 0$, and if S is ruled with irregularity $>0$, then $\chi ({\mathcal {O}}_S)\ge \frac{1}{8}K^2_S$. Therefore, taking into account previous remark, when $d>35$, Theorem 1.1 follows from [6, Theorem 1.1]. In order to examine the range $25<d\le 35$, we are going to refine the line of the argument in the proof of [6, Theorem 1.1].

(iii) When $d=2\delta$ is even, then $\frac{1}{8}d(d-6)+1$ is the genus of a plane curve of degree $\delta$, and the genus of a curve of degree d lying on the Veronese surface.

Put $r+1:=h^0(S,{\mathcal {L}})$. Therefore, $|H^0(S,{\mathcal {L}})|$ embeds S in ${\mathbb {P}}^r$. Let $H\subseteq {\mathbb {P}}^{r-1}$ be a general hyperplane section of S, so that ${\mathcal {L}}\cong \mathcal O_S(H)$. We denote by g the genus of H. If $2\le r\le 3$, then $\chi ({\mathcal {O}}_S)\ge 1$. Therefore, we may assume $r\ge 4$.

The case $r=4$.

We first examine the case $r=4$. In this case we only have to prove that, for $d>25$, one has $\chi ({\mathcal {O}}_S)> -\frac{1}{8}d(d-6)$. We may assume that S is an irrational ruled surface, so $K^2_S\le 8\chi ({\mathcal {O}}_S)$ (compare with previous Remark 2.1, (ii)). We argue by contradiction, and assume also that

$$\begin{aligned} \chi ({\mathcal {O}}_S)\le -\frac{1}{8}d(d-6). \end{aligned}$$

(1)

We are going to prove that this assumption implies $d\le 25$, in contrast with our hypothesis $d>25$.

By the double point formula:

$$\begin{aligned} d(d-5)-10(g-1)+12\chi ({\mathcal {O}}_S)=2K^2_S, \end{aligned}$$

and $K^2_S\le 8\chi ({\mathcal {O}}_S)$, we get:

$$\begin{aligned} d(d-5)-10(g-1)\le 4\chi ({\mathcal {O}}_S). \end{aligned}$$

And from $\chi ({\mathcal {O}}_S)\le -\frac{1}{8}d(d-6)$ we obtain

$$\begin{aligned} 10g\ge \frac{3}{2}d^2-8d+10. \end{aligned}$$

(2)

Now we distinguish two cases, according that S is not contained in a hypersurface of degree $<5$ or not.

First suppose that S is not contained in a hypersurface of ${\mathbb {P}}^4$ of degree $<5$. Since $d > 16$, by Roth’s Theorem ([12, p. 152], [8, p. 2, (C)]), H is not contained in a surface of ${\mathbb {P}}^3$ of degree $<5$. Using Halphen’s bound [9], we deduce that

$$\begin{aligned} g\le \frac{d^2}{10} + \frac{d}{2}+1-\frac{2}{5}(\epsilon +1)(4-\epsilon ), \end{aligned}$$

where $d-1=5m+\epsilon$, $0\le \epsilon <5$. It follows that

$$\begin{aligned} \frac{3}{2}d^2-8d+10\le \,10 g\,\le d^2+5d+10\left( 1-\frac{2}{5}(\epsilon +1)(4-\epsilon )\right) . \end{aligned}$$

This implies that $d\le 25$, in contrast with our hypothesis $d>25$.

In the second case, assume that S is contained in an irreducible and reduced hypersurface of degree $s\le 4$. When $s\in \{2,3\}$, one knows that, for $d>12$, S is of general type [2, p. 213]. Therefore, we only have to examine the case $s=4$. In this case H is contained in a surface of ${\mathbb {P}}^3$ of degree 4. Since $d>12$, by Bezout’s Theorem, H is not contained in a surface of ${\mathbb {P}}^3$ of degree $<4$. Using Halphen’s bound [9], and [8, Lemme 1], we get:

$$\begin{aligned} \frac{d^2}{8}-\frac{9d}{8}+1\le \, g\,\le \frac{d^2}{8}+1. \end{aligned}$$

Hence, there exists a rational number $0\le x\le 9$ such that

$$\begin{aligned} g=\frac{d^2}{8}+d\left( \frac{x-9}{8}\right) +1. \end{aligned}$$

If $0\le x\le \frac{15}{2}$, then $g\le \frac{d^2}{8}-\frac{3}{16}d+1$, and from (2) we get

$$\begin{aligned} \frac{3}{20}d^2-\frac{4}{5}d+1\,\le g\,\le \frac{d^2}{8}-\frac{3}{16}d+1. \end{aligned}$$

It follows $d\le 24$, in contrast with our hypothesis $d>25$.

Assume $\frac{15}{2}< x\le 9$. Hence,

$$\begin{aligned} \left( \frac{d^2}{8}+1\right) -g= -d\left( \frac{x-9}{8}\right) <\frac{3}{16}d. \end{aligned}$$

By [5, proof of Proposition 2, and formula (2.2)], we have

$$\begin{aligned} \chi ({\mathcal {O}}_S)\ge & {} 1+ \frac{d^3}{96}-\frac{d^2}{16}-\frac{5d}{3}-\frac{349}{16}-(d-3) \left[ \left( \frac{d^2}{8}+1\right) -g\right] \\> & {} 1+ \frac{d^3}{96}-\frac{d^2}{16}-\frac{5d}{3}-\frac{349}{16}-(d-3)\frac{3}{16}d = \frac{d^3}{96}-\frac{d^2}{4}-\frac{53}{48}d-\frac{333}{16}. \end{aligned}$$

Combining with (1), we get

$$\begin{aligned} \frac{d^3}{96}-\frac{d^2}{4}-\frac{53}{48}d- \frac{333}{16}+\frac{1}{8}d(d-6)<0, \end{aligned}$$

i.e.

$$\begin{aligned} d^3-12d^2-178d-1998<0. \end{aligned}$$

It follows $d\le 23$, in contrast with our hypothesis $d>25$.

This concludes the analysis of the case $r=4$.

The case $r\ge 5$.

When $r\ge 5$, by [6, Remark 2.1], we know that, for $d>5$, one has $K^2_S>-d(d-6)$, except when $r=5$, and the surface S is a scroll, $K^2_S=8\chi ({\mathcal {O}}_S)=8(1-g)$, and

$$\begin{aligned} g=\frac{1}{8}d^2-\frac{3}{4}d+\frac{(5-\epsilon )(\epsilon +1)}{8}, \end{aligned}$$

(3)

with $d-1=4m+\epsilon$, $0<\epsilon \le 3$. In this case, by [6, pp. 73–76], we know that, for $d>30$, S is contained in a smooth rational normal scroll of ${\mathbb {P}}^5$ of dimension 3. Taking into account that we may assume $K^2_S\le 8\chi (\mathcal O_S)$ (compare with Remark 2.1, (i) and (ii)), at this point Theorem 1.1 follows from [6, Proposition 2.2], when $d>30$.

In order to examine the remaining cases $26\le d \le 30$, we refine the analysis appearing in [6]. In fact assuming that $r=5$ and S is a scroll, and assuming that (3) holds, then S is contained in a smooth rational normal scroll of $\mathbb P^5$ also in the range $26\le d \le 30$. Then we may conclude as before, because [6, Proposition 2.2] holds true for $d\ge 18$.

First, observe that if S is contained in a threefold $T\subset {\mathbb {P}}^5$ of dimension 3 and minimal degree 3, then T is necessarily a smooth rational normal scroll [6, p. 76]. Moreover, observe that we may apply the same argument as in [6, pp. 75–76] in order to exclude the case S is contained in a threefold of degree 4. In fact the argument works for $d>24$ [6, p. 76, first line after formula (13)].

In conclusion, assuming that $r=5$ and S is a scroll, and assuming that (3) holds, it remains to exclude that S is not contained in a threefold of degree $<5$ in the range $26\le d \le 30$.

Assume S is not contained in a threefold of degree $<5$. Denote by $\Gamma \subset {\mathbb {P}}^3$ a general hyperplane section of H. Recall that $26\le d \le 30$.

$\bullet$ Case I $h^0({\mathbb {P}}^3,\mathcal I_{\Gamma }(2))\ge 2$.

It is impossible. In fact, if $d>4$, by monodromy [4, Proposition 2.1], $\Gamma$ should be contained in a reduced and irreducible space curve of degree $\le 4$, and so, for $d>20$, S should be contained in a threefold of degree $\le 4$ [3, Theorem (0.2)].

$\bullet$ Case II $h^0({\mathbb {P}}^3,\mathcal I_{\Gamma }(2))=1$ and $h^0({\mathbb {P}}^3,{\mathcal {I}}_{\Gamma }(3))>4$.

As before, if $d>6$, by monodromy, $\Gamma$ is contained in a reduced and irreducible space curve X of degree $\deg (X)\le 6$. Again as before, if $\deg (X)\le 4$, then S is contained in a threefold of degree $\le 4$. So we may assume $5\le \deg (X)\le 6$.

Denote by $h_{\Gamma }$ and $h_X$ the Hilbert function of $\Gamma$ and X. First notice that, since $\Gamma \subset {\mathbb {P}}^3$ is non degenerate, and $h^0({\mathbb {P}}^3,{\mathcal {I}}_{\Gamma }(2))=1$, we have:

$$\begin{aligned} h_{\Gamma }(1)=4, \quad {\text {and}} \quad h_{\Gamma }(2)=9. \end{aligned}$$

(4)

Moreover, since $d\ge 26$, by Bezout’s Theorem we have

$$\begin{aligned} h_{\Gamma }(i)=h_X(i) \quad {\text {for every}} \quad i\le 4. \end{aligned}$$

(5)

Let $X'\subset {\mathbb {P}}^2$ be a general plane section of X, and $h_{X'}$ its Hilbert function. By [7, Lemma (3.1), p. 83] we know that $h_X(i)-h_X(i-1)\ge h_{X'}(i)$ for every i. Therefore, for every i, we have:

$$\begin{aligned} h_X(i)\ge \sum _{j=0}^{i}h_{X'}(j). \end{aligned}$$

(6)

On the other hand, by [7, Corollary (3.6), p. 87], we also know that

$$\begin{aligned} h_{X'}(j)\ge \min \{2j+1,\, \deg (X)\}. \end{aligned}$$

(7)

Therefore, by (5), (6), and (7) (recall that $5\le \deg (X)\le 6$), we get:

$$\begin{aligned} h_{\Gamma }(3)\ge 14 \quad {\text {and}} \quad h_{\Gamma }(4)\ge 19. \end{aligned}$$

(8)

By [7, Corollary (3.5), p. 86] we have:

$$\begin{aligned} h_{\Gamma }(i+j)\ge \min \{d,\, h_{\Gamma }(i)+h_{\Gamma }(j)-1 \} \quad {\hbox {for every}\,\, i\,\, \hbox {and}\,\, j}. \end{aligned}$$

(9)

Combining (9) with (4) and (8), we get:

$$\begin{aligned} h_{\Gamma }(5)\ge 22, \, h_{\Gamma }(6)\ge \min \{d,\, 27\},\, h_{\Gamma }(7)=d. \end{aligned}$$

(10)

Since in general we have [7, Corollary (3.2) p. 84]

$$\begin{aligned} g\le \sum _{i=1}^{+\infty }d-h_{\Gamma }(i), \end{aligned}$$

(11)

from (4), (8), and (10), taking into account that $26\le d\le 30$, it follows that:

$$\begin{aligned} g\le (d-4)+(d-9)+(d-14)+(d-19)+(d-22)+3=5d-65, \end{aligned}$$

which is $<\frac{1}{8}d(d-6)+1$ for $d \ge 26$. This is in contrast with (3).

$\bullet$ Case III $h^0({\mathbb {P}}^3,\mathcal I_{\Gamma }(2))=1$ and $h^0({\mathbb {P}}^3,{\mathcal {I}}_{\Gamma }(3))=4$.

Using these assumptions, by (4) and (9), we have:

$$\begin{aligned} h_{\Gamma }(1)=4,\, h_{\Gamma }(2)=9,\, h_{\Gamma }(3)=16, \, h_{\Gamma }(4)\ge 19, h_{\Gamma }(5)\ge 24, \, h_{\Gamma }(6)=d. \end{aligned}$$

By (11) it follows that:

$$\begin{aligned} g\le (d-4)+(d-9)+(d-16)+(d-19)+(d-24)=5d-72, \end{aligned}$$

which is $< \frac{1}{8}d(d-6)+1$ for $d \ge 26$. This is in contrast with (3).

$\bullet$ Case IV $h^0({\mathbb {P}}^3,\mathcal I_{\Gamma }(2))=0$.

Using this assumption, by (4) and (9), we have:

$$\begin{aligned}&h_{\Gamma }(1)=4,\, h_{\Gamma }(2)=10,\, h_{\Gamma }(3)\ge 13, \, h_{\Gamma }(4)\ge 19, \\&h_{\Gamma }(5)\ge 22,\, h_{\Gamma }(6)\ge \min \{d,\, 28\}, \, h_{\Gamma }(7)=d. \end{aligned}$$

By (11) it follows that:

$$\begin{aligned} g\le (d-4)+(d-10)+(d-13)+(d-19)+(d-22)+2=5d-66, \end{aligned}$$

which is $< \frac{1}{8}d(d-6)+1$ for $d \ge 26$. This is in contrast with (3).

This concludes the proof of Theorem 1.1.

Remark 2.2

(i) Let $Q\subseteq {\mathbb {P}}^3$ be a smooth quadric, and $H\in |{\mathcal {O}}_Q(1,d-1)|$ be a smooth rational curve of degree d [11, p. 231, Exercise 5.6]. Let $S\subseteq {\mathbb {P}}^4$ be the projective cone over H. A computation, which we omit, proves that

$$\begin{aligned} \chi ({\mathcal {O}}_S)=1-\left( {\begin{array}{c}d-1\\ 3\end{array}}\right) . \end{aligned}$$

Therefore, if S is singular, it may happen that $\chi (\mathcal O_S)<-\frac{1}{8}d(d-6)$. One may ask whether $1-\left( {\begin{array}{c}d-1\\ 3\end{array}}\right)$ is a lower bound for $\chi ({\mathcal {O}}_S)$ for every integral surface.

(ii) Let $(S,{\mathcal {L}})$ be a smooth surface, polarized by a very ample line bundle ${\mathcal {L}}$ of degree d. By Harris’ bound for the geometric genus $p_g(S)$ of S [10], we see that $p_g(S)\le \left( {\begin{array}{c}d-1\\ 3\end{array}}\right)$. Taking into account that for a smooth surface one has $\chi ({\mathcal {O}}_S)=h^0(S,\mathcal O_S)-h^1(S,{\mathcal {O}}_S)+h^2(S,{\mathcal {O}}_S) \le 1+h^2(S,\mathcal O_S)=1+p_g(S)$, from Theorem 1.1 we deduce (the first inequality only when $d>25$):

$$\begin{aligned} -\left( {\begin{array}{c}\frac{d}{2}-1\\ 2\end{array}}\right) \le \chi ({\mathcal {O}}_S)\le 1+\left( {\begin{array}{c}d-1\\ 3\end{array}}\right) . \end{aligned}$$

3 Proof of Corollary 1.2

$\bullet$ First, assume that $\chi (\mathcal O_S)=-\frac{1}{8}d(d-6)$.

By Theorem 1.1, we know that $r=5$. Moreover, S is contained in a nonsingular threefold $T\subseteq {\mathbb {P}}^5$ of minimal degree 3. Therefore, a general hyperplane section $H=S\cap L$ of S ($L\cong {\mathbb {P}}^4$ denotes a general hyperplane of ${\mathbb {P}}^5$) is contained in a smooth surface $\Sigma =T\cap L$ of $L\cong {\mathbb {P}}^4$, of minimal degree 3.

This surface $\Sigma$ is isomorphic to the blowing-up of $\mathbb P^2$ at one point [1, p. 58]. Moreover, if V denotes the Veronese surface in ${\mathbb {P}}^5$, for a suitable point $x\in V\backslash L$, the projection of ${\mathbb {P}}^5\backslash \{x\}$ on $L\cong {\mathbb {P}}^4$ from x restricts to an isomorphism

$$\begin{aligned} p_x:V\backslash \{x\}\rightarrow \Sigma \backslash E, \end{aligned}$$

where E denotes the exceptional line of $\Sigma$ [1, loc. cit.].

Since S is linearly equivalent on T to $\frac{d}{2}(H_T- W_T)$ ($H_T$ denotes the hyperplane section of T, and $W_T$ the ruling), it follows that H is linearly equivalent on $\Sigma$ to $\frac{d}{2}(H_{\Sigma }- W_{\Sigma })$ (now $H_{\Sigma }$ denotes the hyperplane section of $\Sigma$, and $W_{\Sigma }$ the ruling of $\Sigma$). Therefore, H does not meet the exceptional line $E=H_{\Sigma }- 2W_{\Sigma }$. In fact, since $H_{\Sigma }^2=3$, $H_{\Sigma }\cdot W_{\Sigma }=1$, and $W_{\Sigma }^2=0$, one has:

$$\begin{aligned} (H_{\Sigma }- W_{\Sigma })\cdot (H_{\Sigma }- 2W_{\Sigma })= H_{\Sigma }^2-3H_{\Sigma }\cdot W_{\Sigma }+2W_{\Sigma }^2=0. \end{aligned}$$

This implies that H is contained in $\Sigma \backslash E$, and the assertion of Corollary 1.2 follows.

$\bullet$ Conversely, assume there exists a curve C on the Veronese surface $V\subseteq {\mathbb {P}}^5$, and a point $x\in V\backslash C$, such that H is the projection $p_x(C)$ of C from the point x.

In particular, d is an even number, and H is contained in a smooth surface $\Sigma \subseteq L\cong {\mathbb {P}}^4$ of minimal degree, and is disjoint from the exceptional line $E\subseteq \Sigma$. By [3, Theorem (0.2)], S is contained in a threefold $T\subseteq {\mathbb {P}}^5$ of minimal degree. T is nonsingular. In fact, otherwise, H should be a Castelnuovo’s curve in ${\mathbb {P}}^4$ [6, p. 76]. On the other hand, by our assumption, H is isomorphic to a plane curve of degree $\frac{d}{2}$. Hence, we should have:

$$\begin{aligned} g=\frac{d^2}{6}-\frac{2}{3}d+1=\frac{d^2}{8}-\frac{3}{4}d+1 \end{aligned}$$

(the first equality because H is Castelnuovo’s, the latter because H is isomorphic to a plane curve of degree $\frac{d}{2}$). This is impossible when $d>0$.

Therefore, S is contained in a smooth threefold T of minimal degree in ${\mathbb {P}}^5$.

Now observe that in $\Sigma$ there are only two families of curves of degree even d and genus $g=\frac{d^2}{8}-\frac{3}{4}d+1$. These are the curves linearly equivalent on $\Sigma$ to $\frac{d}{2}(H_{\Sigma }- W_{\Sigma })$, and the curves equivalent to $\frac{d+2}{6}H_{\Sigma }+ \frac{d-2}{2}W_{\Sigma }$. But only in the first family the curves do not meet E. Hence, H is linearly equivalent on $\Sigma$ to $\frac{d}{2}(H_{\Sigma }- W_{\Sigma })$. Since the restriction ${\text {Pic}}(T)\rightarrow {\text {Pic}}(\Sigma )$ is bijective, it follows that S is linearly equivalent on T to $\frac{d}{2}(H_{T}- W_{T})$. By Theorem 1.1, S is a fortiori linearly normal, and of minimal Euler characteristic $\chi ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$.