Secant loci on moduli of Prym varieties

TL;DR

The work extends Brill-Noether and secant-locus methods from curves on K3 surfaces to Prym curves, establishing that for a general Prym pair [C,η] of genus g, the Prym-canonical secant loci V_e^{e-f}(ω_C⊗η) have the expected dimension dim = e−f(g−1−e+f) and vanish when negative. It then leverages Nikulin surfaces (standard and non-standard) to produce explicit Prym curves disjoint from the Prym difference divisor 𝔇_{2i+1}, enabling precise intersection calculations and the determination of the divisor class of 𝔇_{2i+2} up to scale; these results feed into the proof that the Prym moduli 𝔯_g is of general type for odd g≥13. An alternative Raynaud-theta divisors approach yields a second proof of nonexistence for certain secants and extends to non-standard Nikulin settings, while Mayer–Vietoris arguments for non-standard Nikulin surfaces provide parallel disjointness results. The paper closes with open questions about arithmetic incarnations, possible Du Val-type Nikulin analogues, and a deeper understanding of the χ-map between Prym and Hurwitz loci in the moduli space.

Abstract

We present a Prym analogue of Lazarsfeld's result that curves on general polarized K3 surfaces verify the Brill-Noether Theorem, or equivalently, that their canonical embedding has no unexpected secants. We show that the Prym-canonical embedding of a curve on a general Nikulin surface (both of standard and non-standard types) has no unexpected secants. We then explain how these geometric facts suffice to determine the class of the universal difference divisor on the moduli space of stable Prym curves of (odd) genus g.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Remark 3.2
• Remark 3.3