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