Date of Graduation

5-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Niu, Wenbo

Committee Member

Bergdall, John

Second Committee Member

Miller, Lance E.

Keywords

Algebraic Geometry; Koszul Cohomology; Secant Bundles; Symmetric Products of Curves; Syzygies

Abstract

In this thesis, we give a complete classification of the Koszul cohomology groups Kp,1(C, B, ωC ⊗ B) on a smooth curve C of genus g ≥ 2 with B p-very ample. Such a classification in the case B = ωC has been incomplete until [5]. The classification follows easily from our main result which precisely calculates Kp,1(C, B, B ωC ⊗ B) in terms of h0(C, B). This result also handles the case B = ωC. A straightforward application of Koszul duality yields a corollary to the main result which completely classifies the Koszul groups Kp,1(C, ω2C) for arbitrary C.

In [6] it was proven that for a line bundle L of sufficiently large degree, Kp,1(C, L) ≠ 0 if and only if p ∈ [1, r(L) − gon C]. The other rows of the Betti diagram are straightforward to calculate when L is nonspecial. The primary technique there re-packaged some of the main arguments of [32] which essentially allow one to trade Koszul vanishing for vanishing of sheaf cohomology of secant bundles on symmetric products Cp. The authors then conjectured that a lower bound guaranteeing this biconditional is linear in g. This was confirmed in [28] where the bound deg L ≥ 4g − 3 was constructed. The technique there involves resolving the kernel of a certain short exact sequence and showing cohomology vanishing by pulling back along the quotient morphism q : Cp → Cp. We show that this argument is essentially repeatable by instead pulling back along σ : Cp × C → C where Cp × C is thought of as the universal divisor associated to the Hilbert scheme C parameterizing the effective divisors of C of degree p + 1. This method proves to be slightly more optimal as it effectively lowers the earlier bound to 4g − 4.

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