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