Invariants of a general branched cover of ${\bf P}^{1}$

Abstract

A. We investigate the resolution of a general branched cover α : C → P 1 in its relative canonical embedding C ⊂ PE. We conjecture that the syzygy bundles appearing in the resolution are balanced for a general cover, provided that the genus is sufficiently large compared to the degree. We prove this for the Casnati-Ekedahl bundle, or bundle of quadrics F-the first bundle appearing in the resolution of the ideal of the relative canonical embedding. Furthermore, we prove the conjecture for all syzygy bundles in the resolution when the genus satisfies g = 1 mod d.

Special codimension one loci in Hurwitz spaces

“…As mentioned already, the question is already interesting for the bundle of quadrics N1, see [BP15] and [BH15].…”

“…The splitting type of F provides an additional discrete invariant of a branched cover which is essentially "independent" of the scrollar invariants, in the sense that one set of invariants does not determine the other. The isomorphism class of F for a generic curve is understood under the provision that g d: The Main Theorem of [BP15] states that F is balanced for a Zariski open set of covers, i.e. Ext 1 (F, F) = 0.…”

“…We have not fully classified the circumstances under which the locus CE is divisorial, but the natural expectation is that this occurs precisely when the rank rk F divides the degree deg F. This translates into the divisibility condition d(d − 3)/2 | (d − 3)(g + d − 1), and it follows from the main theorem of [BP15] that for such pairs (g, d) the CE-locus is an effective divisor. As in the case of M, it is an interesting problem to establish in these instances whether the divisor CE is rigid or extremal -we establish this when d ≤ 5 in Theorem B.…”

“…Very little is known about the Casnati-Ekedahl loci C(F) -a general theorem analogous to part (4) of Theorem 1.9 has only recently been shown in [BP15] using a somewhat delicate degeneration argument. We provide here a simpler proof in the case d ≤ 5 because we will need to refer to the method of proof later.…”

Syzygy divisors on Hurwitz spaces

“…Evidence for this is given by the next theorem. [3]). The bundle N 1 is balanced for a general branched cover provided g is much larger than d. When d divides g − 1, all syzygy bundles N i are balanced for a general branched cover.…”