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We construct Eagon–Northcott cycles on Hurwitz space and compare their classes to Kleiman's multiple point loci. Applying this construction towards the classification of Betti tables of canonical curves, we find that the value of the extremal Betti number records the number of minimal pencils. The result holds under transversality hypotheses equivalent to the virtual cycles having a geometric interpretation. We analyze the case of two minimal pencils, showing that the transversality hypotheses hold generically. [ABSTRACT FROM AUTHOR]

In this article we consider moduli properties of singular curves on K3 surfaces. Let B g denote the stack of primitively polarized K3 surfaces ( X , L ) of genus g and let T g , k n → B g be the stack parameterizing tuples [ ( f : C → X , L ) ] with f an unramified morphism which is birational onto its image, C a smooth curve of genus p ( g , k ) − n and f ⁎ C ∈ | k L | . We show that the forgetful morphism η : T g , k n → M p ( g , k ) − n is generically finite on at least one component, for all but finitely many values of p ( g , k ) − n . We further study the Brill–Noether theory of those curves parametrized by the image of η , and find a Wahl-type obstruction for a smooth curve with an unordered marking to have a nodal model on a K3 surface in such a way that the marking is the divisor over the nodes. [ABSTRACT FROM AUTHOR]