Michael Kemeny

The generic Green–Lazarsfeld Secant Conjecture

Using lattice theory on special

The Prym–Green conjecture for torsion line bundles of high order

K3

The moduli of singular curves on K3 surfaces

K3 surfaces, calculations on moduli stacks of pointed curves and Voisin’s proof of Green’s Conjecture on syzygies of canonical curves, we prove the Prym–Green Conjecture on the naturality of the resolution of a general Prym-canonical curve of odd genus, as well as (many cases of) the Green–Lazarsfeld Secant Conjecture on syzygies of non-special line bundles on general curves.

Stable Maps and Chow Groups

We investigate the universal Severi variety of rational curves on K3 surfaces, which parametrises irreducible rational curves in a fixed class on varying K3 surfaces of fixed genus. We investigate the conjecuted irreducibility of this space in the case of primitive classes using the deformation theory of stable maps and obtain partial results. In an appendix we give a short proof of the irreducibility of universal Severi varieties of high genus curves on K3 surfaces.

Syzygies of curves beyond Green's Conjecture

The Prym-Green Conjecture predicts that the resolution of a generic n-torsion paracanonical curve of every genus is natural. The conjecture has mostly been studied so far for level 2, that is, for Prym-canonical curves. Using a construction of Barth and Verra that realizes torsion bundles on sections of special K3 surfaces, we prove the Prym-Green Conjecture for curves of odd genus g and torsion bundles of sufficiently high order with respect to g. We also give partial results in even genus. In the process, we confirm the expectation of Barth and Verra concerning the number of curves in a fixed linear system on a K3 surface, having an n-torsion line bundle induced by restriction from the K3 surface.

The resolution of paracanonical curves of odd genus

Let

Linear syzygies on curves with prescribed gonality

C

The Classification of Betti tables of Canonical Curves and Hurwitz Space.

be a curve and