Details
| Original language | English |
|---|---|
| Article number | 2150054 |
| Number of pages | 28 |
| Journal | International Journal of Mathematics |
| Volume | 32 |
| Issue number | 8 |
| Early online date | 18 Jun 2021 |
| Publication status | Published - Jul 2021 |
Abstract
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.
Keywords
- abelian scheme, abelian variety, Kummer variety, moduli
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: International Journal of Mathematics, Vol. 32, No. 8, 2150054, 07.2021.
Research output: Contribution to journal › Article › Research › peer review
}
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 -