arXiv subjects

Radu Laza

Publications and source records attributed to Radu Laza.

42 records · Page 3Linked to original sources

arXiv2012

Simultaneous semi-stable reduction for curves with ADE singularities

A key tool in the study of algebraic surfaces and their moduli is Brieskorn's simultaneous resolution for families of algebraic surfaces with simple (du Val or ADE) singularities. In this paper we show that a similar statement holds for families of curves with at worst simple (ADE) singularities. For a family $\mathscr X\to B$ of ADE curves, we give an explicit and natural resolution of the rational map $B\to \bar M_g$. Moreover, we discuss a lifting of this map to the moduli stack $ \bar {\mathcal M}_g$, i.e. a simultaneous semi-stable reduction for the family $\mathscr X/B$. In particular, we note that in contrast to what might be expected from the case of surfaces, the natural Weyl cover of $B$ is not a sufficient base change for a lifting of the map $B\to \bar M_g$ to $\bar {\mathcal M}_g$.

math.AG↗

arXiv2012

The geometry of the ball quotient model of the moduli space of genus four curves

S. Kondo has constructed a ball quotient compactification for the moduli space of non-hyperelliptic genus four curves. In this paper, we show that this space essentially coincides with a GIT quotient of the Chow variety of canonically embedded genus four curves. More specifically, we give an explicit description of this GIT quotient, and show that the birational map from this space to Kondo's space is resolved by the blow-up of a single point. This provides a modular interpretation of the points in the boundary of Kondo's space. Connections with the slope nine space in the Hassett-Keel program are also discussed.

math.AG↗

arXiv2008

Triangulations of the sphere and degenerations of K3 surfaces

W. Thurston proved that to a triangulation of the sphere of non-negative combinatorial curvature, one can associate an element in a certain lattice over the Eisenstein integers such that its orbit is a complete invariant of the triangulation. In this paper, we show that this association can be obtained naturally by using Type III degenerations of K3 surfaces.

math.AG↗

arXiv2008

The moduli space of cubic fourfolds

We describe the GIT compactification of the moduli space of cubic fourfolds, with a special emphasis on the role played by singularities. Our main result is that a cubic fourfold with only isolated simple (A-D-E) singularities is GIT stable. Conversely, with some minor exceptions, the stability for cubic fourfolds is characterized by this condition.

math.AG↗

arXiv2007

Counting the hyperplane sections with fixed invariants of a plane quintic

We use three different methods to count the number of lines in the plane whose intersection with a fixed general quintic has fixed cross-ratios. We compare and contrast these methods, shedding light on some classical ideas which underly modern techniques.

math.AG↗

arXiv2007

Deformations of singularities and variation of GIT quotients

We study the deformations of the minimally elliptic surface singularity $N_{16}$. A standard argument reduces the study of the deformations of $N_{16}$ to the study of the moduli space of pairs $(C,L)$ consisting of a plane quintic curve and a line. We construct this moduli space in two ways: via the periods of $K3$ surfaces and by using geometric invariant theory (GIT). The GIT construction depends on the choice of the linearization. In particular, for one choice of linearization we recover the space constructed via $K3$ surfaces and for another we obtain the full deformation space of $N_{16}$. The two spaces are related by a series of explicit flips. In conclusion, by using the flexibility given by GIT and the standard tools of Hodge theory, we obtain a good understanding of the deformations of $N_{16}$.

math.AG↗