Related papers: Mori's program for the moduli space of pointed sta…

We compactify the moduli stack of maps from curves to certain quotient stacks $\mathcal{X}=[W/G]$ with a projective good moduli space, extending previous results from quasimap theory. For doing so, we introduce a new birational…

We complete Mori's program with symmetric divisors for the moduli space of stable seven pointed rational curves. We describe all birational models in terms of explicit blow-ups and blow-downs. We also give a moduli theoretic description of…

Let $M(n,\xi)$ be the moduli space of stable vector bundles of rank $n\geq 3$ and fixed determinant $\xi$ over a smooth projective algebraic curve $X$ over $\mathbb{C}$ of genus $g\geq 4.$ We use the gonality of the curve and $r$-Hecke…

The stable rationality of components of the moduli space of (unparametrized) rational curves in projective $n$-space with fixed normal bundle is proved, provided these components dominate the moduli space of immersed rational curves in the…

We prove that if $m \ge n-3$ then every $S_m$-invariant F-nef divisor on the moduli space of stable $n$-pointed curves of genus zero is linearly equivalent to an effective combination of boundary divisors. As an application, we determine…

Let X be a smooth projective complex curve, and let M be the moduli space of stable Higgs bundles on X (with genus g>1), with rank n and fixed determinant \xi, with n and deg(\xi) coprime. Let X' and \xi' be another such curve and line…

In the last years the biregular automorphisms of the Deligne-Mumford's and Hassett's compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a…

We construct moduli spaces of linear self-maps of projective space with marked points, up to projective equivalence. That is, we let the special linear group act simultaneously by conjugation on projective linear maps and diagonally on…

In this paper, we completely work out the log minimal model program for the moduli space of stable curves of genus three. We employ a rational multiple $\alpha\delta$ of the divisor $\delta$ of singular curves as the boundary divisor,…

We compute the intersection Betti numbers of the GIT model of the moduli space of Brill-Noether-Petri general curves of genus 4. This space was shown to be the final non-trivial log canonical model for the moduli space of stable genus four…

The purpose of this article is to give an overview of the construction of moduli spaces of curves from the viewpoint of the log minimal model program for M_g by providing an update of recent developments and discussing future problems. This…

In this paper, we study the birational geometry of the Quot schemes of trivial bundles on $\mathbb{P}^1$ by constructing small $\mathbb{Q}$-factorial modifications of the Quot schemes as suitable moduli spaces. We determine all the models…

In this paper we build bridges between moduli theory of sheaf stable pairs on one hand and birational geometry on the other hand. We will in particular treat moduli of sheaf stable pairs on smooth projective curves in detail and present…

Let $C$ be a smooth projective curve of genus $g \geq 2$ over $\mathbb C$, and let $E^0$ be a vector bundle on $C$. We investigate the birational geometry of the Quot scheme ${\rm Quot}_C(E^0, k, n)$, which parametrizes quotients of $E^0$…

We study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n//SL2$. Our main result is that $\overline{M}_{0,n}$ admits a morphism to each such GIT quotient, analogous to the well-known result of…

Let $C$ be a smooth, projective, geometrically irreducible curve defined over $\mathbb{R}$ such that $C(\mathbb{R}) = \emptyset$. Let $r>0$ and $d$ be integers which are coprime. Let $L$ be a line bundle on $C$ which corresponds to an…

Using Moriwaki's calculation of the Q-Picard group for the moduli space of curves, I prove the strong Franchetta Conjecture in all characteristics. That is, the canonical class generates the group of rational points on the Picard scheme for…

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion…

Fulton's conjecture for the moduli space of stable pointed rational curves, \bar{M}_{0,n}, claims that a divisor non-negatively intersecting all F-curves is linearly equivalent to an effective sum of boundary divisors. Our main result is a…

We prove that the moduli spaces of framed bundles over a smooth projective curve are rational. We compute the Brauer group of these moduli spaces to be zero under some assumption on the stability parameter.