We prove that the rational Picard group of the simple Hurwitz space ${\mathcal H}_{d,g}$ is trivial for $d$ up to five. We also relate the rational Picard groups of the Hurwitz spaces to the rational Picard groups of the Severi varieties of nodal curves on Hirzebruch surfaces.Comment: Added a figure. Fixed typos and numerical mistake

We describe a sequence of effective divisors on the Hurwitz space H d,g for d dividing g − 1 and compute their cycle classes on a partial compactification. These divisors arise from vector bundles of syzygies canonically associated to a branched cover. We find that the cycle classes are all proportional to each other.

Abstract. We establish sharp bounds for the slopes of curves in M g that sweep out the locus of trigonal curves, reproving Stankova-Frenkel's bound of 7 + 6/g for even g and obtaining the bound 7 + 20/(3g + 1) for odd g. For even g, we find an explicit expression of the so-called Maroni divisor in the Picard group of the space of admissible triple covers. For odd g, we describe the analogous extremal effective divisor and give a similar explicit expression.

We investigate the resolution of a general branched cover $\alpha \colon C \to \mathbf{P}^1$ in its relative canonical embedding $C \subset \mathbf{P} E$. 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 1st 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 $g = 1 \mod d$.

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