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.
Highlights
The construction of compact moduli spaces with geometric meanings is an important problem in algebraic geometry
We discuss the case of the moduli of K 3 surfaces of degree 2 obtained as minimal resolutions of double covers of P2 branched at a quartic C and two lines L1, L2, for which we give two constructions, one via Geometric Invariant Theory (GIT) for the plane curves (C, L1, L2) depending on a choice of two parameters for each of the lines, and one via the period map of K 3 surfaces
The moduli of del Pezzo surfaces of degree 2 with two anti-canonical sections seems to be closely related to the moduli of K 3 surfaces considered in this article, since del Pezzo surfaces of degree 2 with canonical singularities can be obtained as double-covers of P2 branched at a quartic curve
Summary
The construction of compact moduli spaces with geometric meanings is an important problem in algebraic geometry. 2. Following the general theory of variations of GIT quotients developed by Dolgachev and Hu [9] and independently by Thaddeus [32], we construct GIT compactifications M(t1, t2) for the moduli space of triples (C, L1, L2) consisting of a smooth plane quartic curve C and two labeled lines L1, L2 in Sect. To compare the GIT compactification and the Baily–Borel compactification we consider a slightly different moduli space M∗ (constructed by taking a quotient of the GIT quotient M(1, 1)) parameterizing triples (C, L , L ) consisting of quartic curves C and unlabeled lines L , L. The study of singularities and incidences lines on quartic curves is a classical topic
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