$${\mathbb P^r}$$ -scrolls arising from Brill–Noether theory and K3-surfaces

Abstract

In this paper we study examples of P r -scrolls defined over primitively polarized K3 surfaces S of genus g, which arise from Brill-Noether theory of the general curve in the primitive linear system on S and from Lazarsfeld's results in [25]. We show that such scrolls form an open dense subset of a component H of their Hilbert scheme; moreover, we study some properties of H (e.g. smoothness, dimensional computation, etc.) just in terms of Bg, the moduli space of such K3's, and Mv(S), the moduli space of semistable torsion-free sheaves of a given rank on S.One of the motivation of this analysis is to try to introducing the use of projective geometry and degeneration techniques in order to studying possible limits of semistable vector-bundles of any rank on a general K3 as well as Brill-Noether theory of vectorbundles on suitable degenerations of projective curves.We conclude the paper by discussing some applications to the Hilbert schemes of geometrically ruled surfaces introduced and studied in [9] and [10]. d (C), and of the expected dimension ρ(g, r, d). In other words, C satisfies Petri's condition.An interesting, independent proof of the previous result is contained in [37]. Lazarsfeld's approach for a proof of Theorem 1.1 uses vector-bundle techniques on S. For reader's convenience, we briefly recall Lazarsfeld's construction and set-up in § 4.Roughly speaking, given a smooth, primitively polarized K3 surface (S, L) of genus g ≥ 3 (i.e. L 2 = 2g − 2), a general (smooth) curve C ∈ |L| and a complete linear series |A| = g r d

Rational Curves on and K3 Surfaces

“…Since the fibres of π g are irreducible, P g is an irreducible stack of dimension 19 + g. The tangent space to P g at a pair (S, C) with C smooth is H 1 (T S C ) (see e.g. [11], Section 3).…”

“…Remark 6.3. Theorem 6.2 shows that, for g = 7, 8,9,11, the morphism Ψ f is a so-called minimal free morphism. Given a projective variety X of dimension n and a free morphism h : P 1 → X, h is said to be minimal if the splitting of h * T X is:…”

On families of rank-2 uniform bundles on Hirzebruch surfaces and Hilbert schemes of their scrolls

“…As for the Palatini scroll, its Hilbert scheme is described in [22], while its natural generalizations as Hilbert schemes of scrolls that arise as degeneracy loci of general morphisms φ : O ⊕m P 2k−1 −→ Ω P 2k−1 (2) are studied in [17] and [18]. Further examples of K3-scrolls are presented in [23]. Turning our attention to rank-two vector bundles over ruled surfaces, one first has the complete classification given by Brosius [10], who introduced a canonical way of representing them as extensions of suitable coherent sheaves as in Segre's approach for curves.…”