## 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 \(x\in V\backslash C\).

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## 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*:

*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

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

By the double point formula:

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

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

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

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

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:

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

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

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

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

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

Combining with (1), we get

i.e.

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

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:

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

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:

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

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

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

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

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

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

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:

By (11) it follows that:

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:

By (11) it follows that:

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

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 \(d>25\)):

## 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

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:

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:

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

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Gennaro, V.D. A lower bound for \(\chi ({\mathcal {O}}_S)\).
*Rend. Circ. Mat. Palermo, II. Ser* **71**, 225–231 (2022). https://doi.org/10.1007/s12215-021-00618-6

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DOI: https://doi.org/10.1007/s12215-021-00618-6