Related papers: Genus-One Stable Maps, Local Equations, and Vakil-…

We study normal crossings compactifications of the moduli space of maps $\mathcal{M}_{g, n}(\mathbb{P}^r, d)$, for $g = 0$ and $g = 1$. In each case we explicitly determine the dual boundary complex, and prove that it admits a natural…

We define two equivalent notions of twisted stable map from a curve to a Deligne-Mumford stack with projective moduli space, and we prove that twisted stable maps of fixed degree form a complete Deligne-Mumford stack with projective moduli…

We consider the stack of stable curves of genus g with a given dual graph and we give an explicit desingularization of its closure in the moduli stack of stable curves. We study in particular the one-dimensional substack of curves with at…

We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of…

The Gieseker-Uhlenbeck morphism maps the Gieseker moduli space of stable rank-2 sheaves on a smooth projective surface to the Uhlenbeck compactification, and is a generalization of the Hilbert-Chow morphism for Hilbert schemes of points.…

The new compactification of moduli scheme of Gieseker-stable vector bundles with the given Hilbert polynomial on a smooth projective polarized surface (S;H), over the field k = \bar k of zero characteristic, is constructed in previous…

We discuss criteria for a stable map of genus two and degree $4$ to the projective plane to be smoothable, as an application of our modular desingularisation of $\overline{\mathcal M}_{2,n}(\mathbb{P}^r,d)^{\text{main}}$ via logarithmic…

The moduli space $\cM_g$ of nonsingular projective curves of genus $g$ is compactified into the moduli $\bcM_g$ of Deligne-Mumford stable curves of genus $g$. We compactify in a similar way the moduli space of abelian varieties by adding…

The moduli space of slope-stable vector bundles on a normal projective variety over an algebraically closed field of characteristic $p\geq 0$ is stratified with respect to the decomposition type. On a smooth projective curve of genus at…

The main aim of this paper is to develop general algebraic and cohomological tools for the study of the local geometry of moduli and parameter spaces in Algebraic Geometry, culminating in the so-called Hitchin (or KZ) (projective)…

We develop a theory of good moduli spaces for derived Artin stacks, which naturally generalizes the classical theory of good moduli spaces introduced by Alper. As such, many of the fundamental results and properties regarding good moduli…

In this paper, we study the moduli space of unipotent Weil-Deligne representations valued in a split reductive group $G$ and characterise which irreducible components are smooth. We apply the smoothness results proved to show that a certain…

This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a…

In this paper we propose the concept of formal desingularizations as a substitute for the resolution of algebraic varieties. Though a usual resolution of algebraic varieties provides more information on the structure of singularities there…

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…

Consider the (formal/analytic/algebraic) map-germs Maps(X,(k^p,o)). Let G be the group of right/contact/left-right transformations. I extend the following (classical) results from the real/complex-analytic case to the case of arbitrary…

The space of smooth genus 0 curves in projective space has a natural smooth compactification: the moduli space of stable maps, which may be seen as the generalization of the classical space of complete conics. In arbitrary genus, no such…

Let $X$ be a smooth complex projective curve of genus $g\geq 3$. Let $\mathbf{M}_2$ be the moduli space of semistable rank $2$ Higgs bundles with trivial determinant over $X$. We construct a desingularization $\mathbf{S}$ of $\mathbf{M}_2$…

We study constructible invariants of the moduli space $\overline{\mathcal{M}}(\boldsymbol{x})$ of stable maps from genus zero curves to $\mathbb{P}^1$, relative to $0$ and $\infty$, with ramification profiles specified by…

Let $X$ be a nonsingular projective surface over an algebraically closed field with characteristic zero, and $H_-$ and $H_+$ ample line bundles on $X$ separated by only one wall of type $(c_1,c_2)$. Suppose the moduli scheme $M(H_-)$ of…