Compactifications of the moduli space of plane quartics and two lines

Patricio Gallardo; Jesus Martinez-Garcia; Zheng Zhang

doi:10.1007/s40879-018-0248-7

Compactifications of the moduli space of plane quartics and two lines

Algebraic Geometry 2019-05-30 v2

Authors: Patricio Gallardo , Jesus Martinez-Garcia , Zheng Zhang

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Abstract

We study the moduli space of triples $(C, L_1, L_2)$ consisting of quartic curves $C$ and lines $L_1$ and $L_2$ . Specifically, we construct and compactify the moduli space in two ways: via geometric invariant theory (GIT) and by using the period map of certain lattice polarized $K3$ surfaces. The GIT construction depends on two parameters $t_1$ and $t_2$ which correspond to the choice of a linearization. For $t_1=t_2=1$ we describe the GIT moduli explicitly and relate it to the construction via $K3$ surfaces.

Keywords

moduli space plane quartics geometric invariant theory K3 surfaces period map

Cite

@article{arxiv.1708.08420,
  title  = {Compactifications of the moduli space of plane quartics and two lines},
  author = {Patricio Gallardo and Jesus Martinez-Garcia and Zheng Zhang},
  journal= {arXiv preprint arXiv:1708.08420},
  year   = {2019}
}

Comments

28 pages, 2 figures. To appear in Eur. J. Math

Compactifications of the moduli space of plane quartics and two lines

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