On the projective normality of Enriques surfaces (with an appendix by Angelo Felice Lopez and Alessandro Verra)
The question of projective normality of linearly normal curves of fixed genus \(g\) and degree at least \(2g +1\) is well known and has been studied by several authors. For surfaces some results have been proved in this direction as for example \textit{L. Ein} and \textit{R. Lazarsfeld} [Invent. Math. 111, 51--67 (1993; Zbl 0814.14040)] for varieties of any dimension and more precise results on syzygies or projective normality of surfaces, as it is the case for example by \textit{D. C. Butler} [J. Differ. Geom. 39, 1--34 (1994; Zbl 0808.14024)] for ruled surfaces, \textit{Y. Homma} [Nat. Sci. Rep. Ochanomizu Univ. 31, 61--73 (1980; Zbl 0486.14001)] for the case of elliptic ruled surfaces, \textit{F. Gallego} and \textit{B. P. Purnaprajna} [J. Algebra 186, 597--625 (1996; Zbl 0948.14029)] on projective normality and syzigies of elliptic ruled surfaces, surfaces of general type and Enriques surfaces, among other authors. As the authors of this paper note for the case of \(K3\) surfaces any linearly normal \(K3\) surface is projectively normal and its ideal is generated by quadrics and cubics. They pose the same question for Enriques surfaces and prove in this note their main result, which is stated as follows: Theorem (1.1). Let \(S \subset \mathbb P^{g-1}\) be a linearly normal smooth irreducible Enriques surface. If \(g=6\) and \({\mathcal O}_S(1)\) gives an embedding in a quadric then \(S\) is \(j\)-normal for \(j\geq 3\) and its homogeneous ideal is generated by quadrics and cubics. If either \(g \geq 7\) or \(g=6\) and \({\mathcal O}_S(1)\) does not give an embedding in a quadric then \(S\) is 3-regular in the sense of Castelnuovo-Mumford. In particular \(S\) is projectively normal and its homogeneous ideal is generated by quadrics and cubics. In theorem 1.4 the authors study the geometry of a curve \(C\) on a smooth Enriques surface, whenever \(C\) has gonality \(k \geq 3\). In corollary 1.10 they show that if \(g\) is the genus of \(C\) and \(e\) the Clifford index of a curve \(C\) on a smooth Enriques surface \(S\) with Clifford dimension at least 2 then they give an upper bound of \(g \) in terms of \(e \) as a rational function. They determine the Clifford dimension or \(e\) of \(C_n\) for a general curve \(C_n \in | C| \) except for the case \(e=3, g=13\) and \(C\) numerically equivalent to \( 2E_1 + 2E_2 + 2E_3\) where \(| 2E_i| \) are genus one pencils with \(E_i \cdot E_j = 1\) for \( i \neq j \). They also prove that \(S\) does not contain smooth plane curves of degree \(d \geq 9 \) and that the general \( C_n \in | C| \) is not a smooth plane curve with \( d \in \{ 7,8 \}\). The paper is organized as follows. In section 2, lemma 2.1 is stated and proved to be used in the proof of theorem 1.4 and the proof of corollary 1.10 is also given in this section. The proof of the main result, theorem 1.1 is given in section 3, for which lemma 3.7 is used together with an appendix to this article by A. F. Lopez and A. Verra which studies the cases \( g=7,8\) of theorem 1.1. The article concludes with remark 3.10, claiming that the proof of theorem 1.1 holds for the cases of normal Enriques surfaces stated precisely there.
scientific article; zbMATH DE number 1837350