Non-special scrolls with general moduli

Abstract

Abstract. In this paper we study smooth, non-special scrolls S of degree d, genus g ≥ 0, with general moduli. In particular, we study the scheme of unisecant curves of a given degree on S. Our approach is mostly based on degeneration techniques.

Brill–Noether theory and non-special scrolls

“…In § 2 we collect standard definitions and properties of scrolls and unisecant curves. In § 3 we recall the results in [5] and in [6]. In §'s 4 and 5 we prove the above-mentioned results of the Brill-Noether theory, whereas § 6 contains the enumerative result.…”

“…In [6] we showed that, if g ≥ 1 and S is general, then F is general in U C (d) (cf. [2] and [6,Theorem 5.5]). We then proved that S is a general ruled surface in the sense of Ghione [12], namely the scheme Div 1,m S parametrizing unisecant curves of given degree m on S behaves as expected (for details, cf.…”

“…[30]), then in [12] and, by the present authors, in [6], where we used degeneration techniques. These techniques, in particular the degeneration of a general scroll in H d,g to the union of a rational normal scroll and g quadrics (cf.…”

“…In this section we will fix notation and general assumptions as in [6]. For non reminded terminology we refer the reader to [21], [27], [31] and [6].…”

“…For any closed point [(C, x)] ∈ M 0 g,1 , its fibre via p (n) 1 is isomorphic to Pic (n) (C). As in [6,Theorem 5.4], let q : U d → M 0 g,1 be the relative moduli stack of degree d, rank-two semistable vector bundles; namely, the fibre of q over [(C, x)] is U C (d). Thus, we have the following natural surjective map over M 0…”