Séminaire de Géométrie Complexe
# Ramified double covers of curves on Nikulin surfaces of non-standard type

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Europe/Paris

salle 207 bâtiment 1R2 (Institut de Mathématiques de Toulouse)
### salle 207 bâtiment 1R2

#### Institut de Mathématiques de Toulouse

Description

Polarized Nikulin surfaces (that is, K3 surfaces endowed with a double cover ramified along 8 disjoint rational curves) are of two types, called standard and non-standard. Nikulin surfaces of standard type have been extensively studied in relation to Prym curves and the birational geometry of their moduli space R_g, primarily by Farkas-Verra. On the other hand, Prym curves lying on Nikulin surfaces of non-standard type are known to be quite special in moduli. I will show that Nikulin surfaces of non-standard type are instead the right environment for studying ramified covers of curves and their moduli spaces R_{g,2n}, where 2n is the number of ramification points. By specialization to curves on Nikulin surfaces of non-standard type, I will show that general double covers of curves ramified at 2, 4 or 6 points are Brill-Noether general; this result is in deep contrast with the étale case, since the étale cover of a general even genus g curve is known to be Brill-Noether special. In the second part of the seminar, I will report on work in progress with Knutsen-Verra on unirational parametrizations of R_{g,2}, R_{g,4}, R_{g,6} in low genera, namely, when a general element of the moduli space is expected to lie on a Nikulin surface of non-standard type.