A lower bound for $\chi (\mathcal O_S)$

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 $\chi (\mathcal O_S)\geq -\frac{1}{8}d(d-6)$. 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 the 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 $x\in V\backslash C$.


Introduction
In [6], one proves 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 L, in terms of its degree d = deg 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 χ(O S ) of S [1, p. 2], in the range d > 25. More precisely, we prove the following: Theorem 1.1. Let (S, L) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle L of degree d > 25. Then: The bound is sharp, and the following properties are equivalent.
(ii) h 0 (S, L) = 6, and the linear system |H 0 (S, L)| embeds S in P 5 as a scroll with sectional genus g = 1 8 d(d − 6) + 1; (iii) h 0 (S, L) = 6, d is even, and the linear system |H 0 (S, L)| embeds S in a smooth rational normal scroll T ⊂ P 5 of dimension 3, and here S is linearly equivalent to 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 ⊂ T ).
By Enriques' classification, one knows that if S is unruled or rational, then χ(O S ) ≥ 0. Hence, Theorem 1.1 essentially concerns irrational ruled surfaces.
In the range d > 35, the family of extremal surfaces for χ(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 ⊆ P r be a nondegenerate, smooth, irreducible, projective, complex surface, of degree d > 25. Let L ⊆ P r be a general hyperplane. Then if and only if r = 5, and there is a curve C in the Veronese surface V ⊆ P 5 and a point x ∈ V \C such that the general hyperplane section S ∩ L of S is the projection p x (C) ⊆ L of C in L ∼ = P 4 , from the point x.
In particular, S ∩ L is not linearly normal, instead S is. (iii) When d = 2δ is even, then 1 8 d(d − 6) + 1 is the genus of a plane curve of degree δ, and the genus of a curve of degree d lying on the Veronese surface.
Put r + 1 := h 0 (S, L). Therefore, |H 0 (S, L)| embeds S in P r . Let H ⊆ P r−1 be the general hyperplane section of S, so that L ∼ = O S (H). We denote by g the genus of H. If 2 ≤ r ≤ 3, then χ(O S ) ≥ 1. Therefore, we may assume r ≥ 4.
The case r = 4.
We first examine the case r = 4. In this case we only have to prove that, for . We may assume that S is an irrational ruled surface, so K 2 S ≤ 8χ(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 ≤ 25, in contrast with our hypothesis d > 25. By the double point formula: Now we distinguish two cases, according that S is not contained in a hypersurface of degree < 5 or not.
In the second case, assume that S is contained in an irreducible and reduced hypersurface of degree s ≤ 4. When s ∈ {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 P 3 of degree 4. Since d > 12, by Bezout's Theorem, H is not contained in a surface of P 3 of degree < 4. Using Halphen's bound [9], and [8, Lemme 1], we get: Hence, there exists a rational number 0 ≤ x ≤ 9 such that If 0 ≤ x ≤ 15 2 , then g ≤ d 2 8 − 3 16 d + 1, and from (2) we get 3 20 It follows d ≤ 24, in contrast with our hypothesis d > 25. Assume 15 2 < x ≤ 9. Hence, By [5, proof of Proposition 2, and formula (2. 2)], we have Combining with (1), we get It follows d ≤ 23, in contrast with our hypothesis d > 25. This concludes the analysis of the case r = 4.
The case r ≥ 5.
When r ≥ 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 = 8χ(O S ) = 8(1 − g), and In order to examine the remaining cases 26 ≤ d ≤ 30, we refine the analysis appearing in [6]. In fact, we are going to prove that, assuming r = 5, S is a scroll, and (3), it follows that S is contained in a smooth rational normal scroll of P 5 of dimension 3 also when 26 ≤ d ≤ 30. Then we may conclude as before, because [6, Proposition 2.2] holds true for d ≥ 18.
First, observe that if S is contained in a threefold T ⊂ 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, p. 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 r = 5, S is a scroll, and (3), it remains to exclude that S is not contained in a threefold of degree < 5, when 26 ≤ d ≤ 30.
Assume S is not contained in a threefold of degree < 5. Denote by Γ ⊂ P 3 the general hyperplane section of H. Recall that 26 ≤ d ≤ 30.