A Compactification of the Space of Plane Curves

We prove that $\frac{2}{d}$, $\frac{2d-3}{(d-1)^2}$, $\frac{2d-1}{d(d-1)}$, $\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest log canonical thresholds of reduced plane curves of degree $d\geqslant 3$, and we describe reduced plane curves of degree $d$ whose log canonical thresholds are these numbers. As an application, we prove that $\frac{2}{d}$, $\frac{2d-3}{(d-1)^2}$, $\frac{2d-1}{d(d-1)}$, $\frac{2d-5}{d^2-3d+1}$ and $\frac{2d-3}{d(d-2)}$ are the smallest...

We construct a good compactification of the variety of irreducible projective plane curves of degree n with d nodes and no other singularities.

In this article we address the problem of computing the dimension of the space of plane curves of degree $d$ with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple $(-1)$-curve. We reformulate this conjecture by explicitly l...

This paper studies the moduli space of stable surfaces of general type. The moduli space component containing the moduli point of a product of smooth curves of general type is proved to be the product of the moduli spaces of the curves, unless the curves have the same genus, in which case it is the symmetric product of the moduli space of curves. This result follows from a study of deformation theory of products and the singularities of products of nodal curves (namely that t...

We prove that the moduli spaces of n-pointed m-stable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of the Hassett-Keel log minimal model program for the moduli space of n-pointed stable curves of genus one.

The compactification $\overline M_{1,3}$ of the Gieseker moduli space of surfaces of general type with $K_X^2 =1 $ and $\chi(X)=3$ in the moduli space of stable surfaces parametrises so-called stable I-surfaces. We classify all such surfaces which are 2-Gorenstein into four types using a mix of algebraic and geometric techniques. We find a new divisor in the closure of the Gieseker component and a new irreducible component of the moduli space.

Let $R_4$ be the space of rational plane curves of degree $4$. In this paper, we obtain a sheaf theoretic compactification of $R_4$ via the space of $\alpha$-semistable pairs on $\mathbb{P}^2$ and its birational relations through wall-crossings of semistable pairs. We obtain the Poincar\'e polynomial of the compactified space.

This survey is devoted to the geography problem of log-surfaces constructed as pairs consisting of a smooth projective surface and a reduced boundary divisor. In the first part we focus on the geography problem for log-surfaces associated with pairs of the form $(\mathbb{P}^{2}, C)$, where $C$ is an arrangement of smooth plane curves admitting ordinary singularities. In particular, we focus on the case where $C$ is an arrangement of smooth rational curves. In the second part,...

We prove that the universal family of polarized K3 surfaces of degree 2 can be extended to a flat family of stable slc pairs $(X,\epsilon R)$ over the toroidal compactification associated to the Coxeter fan. One-parameter degenerations of K3 surfaces in this family are described by integral-affine structures on a sphere with 24 singularities.

We classify log-canonical pairs $(X, \Delta)$ of dimension two with $K_X+\Delta$ an ample Cartier divisor with $(K_X+\Delta)^2=1$, giving some applications to stable surfaces with $K^2=1$. A rough classification is also given in the case $\Delta=0$.