The KSBA compactification for the moduli space of degree two K 3 pairs

Radu Laza

Journal of the European Mathematical Society (2016)

  • Volume: 018, Issue: 2, page 225-279
  • ISSN: 1435-9855

Abstract

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Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs ( X , H ) consisting of a degree two K 3 surface X and an ample divisor H . Specifically, we construct and describe explicitly a geometric compactification P ¯ 2 for the moduli of degree two K 3 pairs. This compactification has a natural forgetful map to the Baily–Borel compactification of the moduli space 2 of degree two K 3 surfaces. Using this map and the modular meaning of P ¯ 2 , we obtain a better understanding of the geometry of the standard compactifications of 2 .

How to cite

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Laza, Radu. "The KSBA compactification for the moduli space of degree two $K3$ pairs." Journal of the European Mathematical Society 018.2 (2016): 225-279. <http://eudml.org/doc/277263>.

@article{Laza2016,
abstract = {Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs $(X,H)$ consisting of a degree two $K3$ surface $X$ and an ample divisor $H$. Specifically, we construct and describe explicitly a geometric compactification $\bar\{P\}_2$ for the moduli of degree two $K3$ pairs. This compactification has a natural forgetful map to the Baily–Borel compactification of the moduli space $\mathcal \{F\}_2$ of degree two $K3$ surfaces. Using this map and the modular meaning of $\bar\{P\}_2$, we obtain a better understanding of the geometry of the standard compactifications of $_2$.},
author = {Laza, Radu},
journal = {Journal of the European Mathematical Society},
keywords = {$K3$ surfaces; moduli space of $K3$ surfaces; KSBA; surfaces; moduli space of surfaces; KSBA},
language = {eng},
number = {2},
pages = {225-279},
publisher = {European Mathematical Society Publishing House},
title = {The KSBA compactification for the moduli space of degree two $K3$ pairs},
url = {http://eudml.org/doc/277263},
volume = {018},
year = {2016},
}

TY - JOUR
AU - Laza, Radu
TI - The KSBA compactification for the moduli space of degree two $K3$ pairs
JO - Journal of the European Mathematical Society
PY - 2016
PB - European Mathematical Society Publishing House
VL - 018
IS - 2
SP - 225
EP - 279
AB - Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs $(X,H)$ consisting of a degree two $K3$ surface $X$ and an ample divisor $H$. Specifically, we construct and describe explicitly a geometric compactification $\bar{P}_2$ for the moduli of degree two $K3$ pairs. This compactification has a natural forgetful map to the Baily–Borel compactification of the moduli space $\mathcal {F}_2$ of degree two $K3$ surfaces. Using this map and the modular meaning of $\bar{P}_2$, we obtain a better understanding of the geometry of the standard compactifications of $_2$.
LA - eng
KW - $K3$ surfaces; moduli space of $K3$ surfaces; KSBA; surfaces; moduli space of surfaces; KSBA
UR - http://eudml.org/doc/277263
ER -

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