Birational Geometry Of Moduli Space Research Articles

Birational geometry of moduli spaces of configurations of points on the line

In this paper we study the geometry of GIT configurations of $n$ ordered points on $\mathbb{P}^1$ both from the the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient $(\mathbb{P}^1)^n//PGL(2)$, taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of $(\mathbb{P}^1)^n//PGL(2)$ in its natural embedding, and its group of automorphisms.

On the birational geometry of spaces of complete forms I: collineations and quadrics

Moduli spaces of complete collineations are wonderful compactifications of spaces of linear maps of maximal rank between two fixed vector spaces. We investigate the birational geometry of moduli spaces of complete collineations and quadrics from the point of view of Mori theory. We compute their effective, nef and movable cones, the generators of their Cox rings, and their groups of pseudo-automorphisms. Furthermore, we give a complete description of both the Mori chamber and stable base locus decompositions of the effective cone of the space of complete collineations of the 3-dimensional projective space.

Birational geometry of moduli spaces of stable objects on Enriques surfaces

Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.

Birational models of moduli spaces of coherent sheaves on the projective plane

We study the birational geometry of moduli spaces of semistable sheaves on the projective plane via Bridgeland stability conditions. We show that the entire MMP of their moduli spaces can be run via wall-crossing. Via a description of the walls, we give a numerical description of their movable cones, along with its chamber decomposition corresponding to minimal models. As an application, we show that for primitive vectors, all birational models corresponding to open chambers in the movable cone are smooth and irreducible.

Rank-two vector bundles on non-minimal ruled surfaces

We continue previous works by various authors and study the birational geometry of moduli spaces of stable rank-two vector bundles on surfaces with Kodaira dimension $-\infty$. To this end, we express vector bundles as natural extensions, by using two numerical invariants associated to vector bundles, similar to the invariants defined by Brinzanescu and Stoia in the case of minimal surfaces. We compute explicitly these natural extensions on blowups of general points on a minimal surface. In the case of rational surfaces, we prove that any irreducible component of a moduli space is either rational or stably rational.

The birational geometry of moduli spaces of sheaves on the projective plane

We describe a close relation between wall crossings in the birational geometry of modulispaceofGiesekerstablesheaves MH (v)on P 2 andmini-wallcrossingsinthestability manifold Stab(D b (P 2 )). The general philosophy is that the birational geometry of MH (v) is closely related to the birational geometry of X.S o if−K X is ample, we hope −K MH (v) is ample as well. This turns out to be not quite right but is close. We first show that if v is a primitive topological type such that MH (v) is non-empty and irreducible, then MH (v) is smooth and −K MH (v) is big and nef. In particular, MH (v) is a Mori dream space. Therefore MH (v) provide a rich class of examples of the so called weak Fano varieties, whose classification theory is highly interesting in its own right. We achieve the statement by showing that the first Chern class

On the birational geometry of moduli spaces of pointed curves

We prove that the moduli space ℳg,n of smooth curves of genus g with n marked points is rational for g = 6 and 1 ≤ n ≤ 8, and it is unirational for g = 8 and 1 ≤ n ≤ 11, g = 10 and 1 ≤ n ≤ 3, g = 12 and n = 1.

Walls for Gieseker semistability and the Mumford-Thaddeus principle for moduli spaces of sheaves over higher dimensional bases

Let X be a projective manifold over \( \Bbb C \). Fix two ample line bundles H0 and H1 on X. It is the aim of this note to study the variation of the moduli spaces of Gieseker semistable sheaves for polarizations lying in the cone spanned by H0 and H1. We attempt a new definition of walls which naturally describes the behaviour of Gieseker semistability. By means of an example, we establish the possibility of non-rational walls which is a substantially new phenomenon compared to the surface case. Using the approach by Ellingsrud and Gottsche via parabolic sheaves, we were able to show that the moduli spaces undergo a sequence of GIT flips while passing a rational wall. We hope that our results will be helpful in the study of the birational geometry of moduli spaces over higher dimensional bases.