TY - JOUR

T1 - Moduli spaces of abstract and embedded Kummer varieties

AU - Galeotti, Mattia

AU - Perna, Sara

N1 - Acknowledgments: We are grateful to Alfio Ragusa, Francesco Russo and Giuseppe Zappal`a for organizing Pragmatic 2015, where this work started. Special thanks are also due toGiulio Codogni and Filippo Viviani for introducing us to the problem and for their support and comments. Finally, many thanks to the anonymous reviewer of the first version of this paper, for his precious suggestions

PY - 2021/7

Y1 - 2021/7

N2 - In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack gabs of abstract Kummer varieties and the second one is the stack gem of embedded Kummer varieties. We will prove that gabs is a Deligne-Mumford stack and its coarse moduli space is isomorphic to Ag, the coarse moduli space of principally polarized abelian varieties of dimension g. On the other hand, we give a modular family g → U of embedded Kummer varieties embedded in 2g-1 × 2g-1, meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space K2em of embedded Kummer surfaces and prove that it is obtained from A2 by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: Kgem could be obtained from Ag via a contraction for all g > 1.

AB - In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack gabs of abstract Kummer varieties and the second one is the stack gem of embedded Kummer varieties. We will prove that gabs is a Deligne-Mumford stack and its coarse moduli space is isomorphic to Ag, the coarse moduli space of principally polarized abelian varieties of dimension g. On the other hand, we give a modular family g → U of embedded Kummer varieties embedded in 2g-1 × 2g-1, meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space K2em of embedded Kummer surfaces and prove that it is obtained from A2 by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: Kgem could be obtained from Ag via a contraction for all g > 1.

KW - abelian scheme

KW - abelian variety

KW - Kummer variety

KW - moduli

UR - http://www.scopus.com/inward/record.url?scp=85108809163&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1806.00267

DO - 10.48550/arXiv.1806.00267

M3 - Article

AN - SCOPUS:85108809163

VL - 32

JO - International Journal of Mathematics

JF - International Journal of Mathematics

SN - 0129-167X

IS - 8

M1 - 2150054

ER -