Projective models of Enriques surfaces in scrolls

Abstract

We study Cossec's φ-function, which is defined for divisors with positive self-intersection on an Enriques surface. In this paper we study the existence of pairs`C 2 , φ(C)´with C an irreducible curve. The φ-function gives in a natural way scrolls containing Enriques surfaces. We compute scroll types to some of these scrolls. IntroductionIn Section 1 we examine the numerical properties of Cossec's φ-function (see Definition 1.1). We are mainly concerned with the existence of pairs C 2 , φ(C) with C an irreducible curve. We get complete results for small values of φ (see Table 1), while Theorems 1.6 and 1.8 give information for some large values of φ. These results improve previous results in [5]. The φ-function is defined purely lattice-theoretically, but it encodes much geometric information (see for example Proposition 1.2 which is due to Cossec). In the last couple of years the φ-function has been used effectively to study divisors on Enriques surfaces. [12,18] studies the relation between the φ-function and k-very ampleness. The relation between the φ-function and birational k-very ampleness will be considered in a forthcoming paper by A. L. Knutsen. The relation between the φ-function and Brill-Noether theory is studied in [9]. [8] uses the φ-function to study projective normality.In Section 2 we consider scrolls containing the projective model of an Enriques surface. The φ-function gives in a natural way elliptic pencils associated to irreducible curves C. Given such an elliptic pencil we construct a rational normal scroll containing the image of the Enriques surface mapped to a projective space by the complete linear system |C|. We show that just by knowing C 2 and φ(C) we can, in most cases, determine the scroll type of this scroll (see Theorem 2.3). For small C 2 our results are listed in Table 5. In recent years much has been written about scrolls containing K3 surfaces (see [2,11,17]). As far as I know the same is not true for scrolls containing Enriques surfaces. Section 2 is an attempt to alter this. Notations and conventionsWe work over the ground field C. Throughout this paper S will be an Enriques surface. Line bundles and divisors are used with little or no distinction. Linear equivalence of divisors is denoted by ∼, and numerical equivalenceGiven two divisors A and B we write A ≥ B when |A − B| = ∅, i.e., h 0 (A − B) > 0. PreliminariesWe mention here some well-known definitions and results that we will need later on. See [3,4,5] for more of the basic theory of Enriques surfaces and their projective models. *

Brill–Noether theory of curves on Enriques surfaces I: the positive cone and gonality

“…and we are in case (ii-a) by Proposition 2.8 and Lemma 2.14.Remark 2.15. Proposition 1 improves the result of Hana[Ha, Thm.1.8].Lemma 2.16. If L ≡ 2D with D 2 = 10 and φ(D) = 3, then µ(L) = 2φ(L) = 12.…”