Half Nikulin surfaces and moduli of Prym curves

TL;DR

This paper investigates the moduli space of Nikulin surfaces and their relation to Prym curves, establishing the injectivity of a key map and revealing deep analogies with K3 surface moduli through novel degeneration techniques.

Contribution

It proves the generic injectivity of the map from Prym moduli to Prym curves for Nikulin surfaces, using a new degeneration approach involving half Nikulin surfaces.

Findings

The map hi_g is generically injective on each irreducible component.

Established analogies between hi_g and the Mukai map m_g.

Introduced a new degeneration method using half Nikulin surfaces.

Abstract

Let F^N_g be the moduli space of polarized Nikulin surfaces (Y,H) of genus g and let P^N_g be the moduli of triples (Y,H,C), with C in |H| a smooth curve. We study the natural map \chi_g:P^N_g -> R_g, where R_g is the moduli space of Prym curves of genus g. We prove that it is generically injective on every irreducible component, with a few exceptions in low genus. This gives a complete picture of the map \chi_g and confirms some striking analogies between it and the Mukai map m_g: P_g ->M_g for moduli of triples (Y,H,C), where (Y,H) is any genus g polarized K3 surface. The proof is by degeneration to boundary points of a partial compactification of F^N_g. These represent the union of two surfaces with four even nodes and effective anticanonical class, which we call half Nikulin surfaces. The use of this degeneration is new with respect to previous techniques.