On curves on K3 surfaces: III
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Archiv der Mathematik
On curves on K3 surfaces: III Gerriet Martens
Abstract. Let C be a smooth curve of degree d0 lying on a smooth surface in projective space. Exploiting a criterion due to I. Reider we study when the divisors in a linear series of degree d ≤ d0 on C are contained in hyperplanes. For a K3 surface S with Picard group Z2 we succeed if S contains a line. Mathematics Subject Classification. Primary 14E25; Secondary 14H51. Keywords. Linear series, Hyperplanes, K3 surfaces.
1. Introduction. In this note we are concerned with the Problem. Find classes of smooth irreducible projective curves C/C of fixed degree d0 in Pr0 (d0 > 2r0 ≥ 4) admitting an integer 0 < δ0 (C) ≤ d0 such that every divisor moving in a base point free linear series L of degree 0 < d ≤ δ0 (C) on C is contained in a hyperplane section of C. (Hence L is obtained just by projection from C ⊂ Pr0 , with center a linear subspace of Pr0 .) Apparently, the first systematic study of this problem, with δ0 (C) = d0 , is in the paper [4] of Ciliberto and Lazarsfeld, regarding smooth complete intersection curves C of degree d0 in 3-space P3 . This work was recently extended to arbitrary r0 ≥ 3 in [8] by reducing it to curves on smooth surfaces in Pr0 with an infinite cyclic Picard group. From [11, Sect. 5] it is known that the curves C of degree d0 > 4deg(S) on a smooth surface S in Pr0 with Picard group Pic(S) ∼ = Z fit into the problem (again with δ0 (C) = d0 ); note that this applies (as is, in fact, done in [11]) to curves on a general K3 surface of degree 2r0 − 2 in Pr0 (r0 ≥ 3). The proof in [8] resp. [11] used results of I. Reider based on Bogomolov’s inequality; in fact, in [16, 2.10 and 2.11], Reider provides a criterion concerning the general question when for a linear series |D| on a smooth curve C lying on a smooth surface S there is a linear series |E1 | on S such that H 0 (C, E1 |C −D) = 0 (i.e., as a part, the divisors in |D| are cut out on C by curves in |E1 |). It
G. Martens
Arch. Math.
is easy [11, Sect. 5] to apply Reider’s criterion to curves on a smooth surface S with Pic(S) ∼ = Z, and (still having in mind our Problem) one may try to extend this to curves on a smooth surface S with Pic(S) ∼ = Z2 . In [14] this is carried out (mutatis mutandis) for curves C of degree d0 on a rational surface scroll in Pr0 , with some δ0 (C) d0 . It is the aim of this note to explore the virtue of Reider’s method for our problem concerning curves on a smooth K3 surface S with Pic(S) ∼ = Z2 . Let r0 S ⊂ P be such a surface, H a hyperplane section of S of degree deg(H) = H 2 = deg(S) = 2r0 − 2, and X some smooth irreducible curve on S of genus 2 g(X) = X2 + 1 such that Pic(S) = Z · H ⊕ Z · X. Since H and X are linearly independent over Z, the Hodge number α := (X · H)2 − X 2 H 2 = deg(X)2 − 4n(g(X) − 1) of S is positive where we set n := r0 − 1 to achieve a notation 2 + 1. Knutsen’s existence theorem as in [9]. In particular, then, g(X) < deg(X) 4n [9, Theorem 1.1] provides a simple (and not very restrictive) criterion for the 2 holds. In
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