Related papers: Genus Two Stable Maps, Local Equations and Modular…

The moduli space of stable relative maps to the projective line combines features of stable maps and admissible covers. We prove all standard Gromov-Witten classes on these moduli spaces of stable relative maps have tautological…

This paper solves the global moduli problem for regular holonomic D-modules with normal crossing singularities on a nonsingular complex projective variety. This is done by introducing a level structure (which gives rise to…

We construct moduli stacks of stable sheaves for surfaces fibered over marked nodal curves by using expanded degenerations. These moduli stacks carry a virtual class and therefore give rise to enumerative invariants. In the case of a…

Drawing on the theory of Minimal Model Program singularities for foliations, we define relative canonical and log-canonical singularities for algebraic stacks with finite generic stabilisers. We show that if a point has log-canonical…

We construct the derived scheme of stable sheaves on a smooth projective variety via derived moduli of finite graded modules over a graded ring. We do this by dividing the derived scheme of actions of Ciocan-Fontanine and Kapranov by a…

We investigate the Brill-Noether theory of rank-two, degree-$d$ stable vector bundles of speciality $3$ on a general $\nu$-gonal curve of genus $g$, $3 \leq \nu < \lfloor \frac{g+3}{2} \rfloor$. Our approach leverages universal extension…

We construct the Mumford-Knudsen space of n pointed stable rational curves by a sequence of explicit blow-ups from the GIT quotient (P^1)^n//SL(2) with respect to the symmetric linearization O(1,...,1). The intermediate blown-up spaces turn…

We prove that in characteristic p>0 the locus of stable curves of p-rank at most f is pure of codimension g-f in the moduli space of stable curves. Then we consider the Prym map and analyze it using tautological classes. We study the locus…

We consider (local) parametrizations of Teichmuller space $T_{g,n}$ (of genus $g$ hyperbolic surfaces with $n$ boundary components) by lengths of $6g-6+3n$ geodesics. We find a large family of suitable sets of $6g-6+3n$ geodesics, each set…

This semi-expository paper discusses the log minimal model program as applied to the moduli space of curves, especially in the case of curves of genus two. Log canonical models for these moduli spaces can often be constructed using the…

The Minimal Model Program offers natural higher-dimensional analogues of stable $n$-pointed curves and maps: stable pairs consisting of a projective variety $X$ of dimension $\ge2$ and a divisor $B$, that should satisfy a few simple…

Using the machinery of etale homotopy theory a' la Artin-Mazur we determine the etale homotopy types of moduli stacks over $\bar{\Q}$ parametrizing families of algebraic curves of genus g greater than 1 endowed with an action of a finite…

We give a complete description of the two-dimensional moduli spaces of stable Higgs bundles of rank 2 over complex projective line with one irregular singular point, having a regular leading-order term, and endowed with a generic compatible…

Let X be a geometrically irreducible smooth projective curve over a field k. We describe the algebra of endomorphisms of indecomposable unstable vector bundles over X of rank 2 and degree d. Fixing some numerical invariants, namely the…

We apply the supergeometric analogue of Artin's algebraicity criteria to prove algebraicity for four moduli problems in supergeometry: supercurves, super Riemann surfaces, stable supercurves, and stable super Riemann surfaces. The…

We define and study the stack ${\mathcal U}^{ns,a}_{g,g}$ of (possibly singular) projective curves of arithmetic genus g with g smooth marked points forming an ample non-special divisor. We define an explicit closed embedding of a natural…

We prove finite generation of the algebra of type A conformal blocks over arbitrary stable curves of any genus. As an application we construct a flat family of irreducible normal projective varieties over the moduli stack of stable pointed…

We study the tautological ring of the moduli space of stable n-pointed curves of genus two with rational tails. The algebra is described in terms of explicit generators and relations. It is proven that this algebra is Gorenstein.

We study the Fulton-Macpherson operational Chow rings of good moduli spaces of properly stable, smooth, Artin stacks. Such spaces are \'etale locally isomorphic to geometric invariant theory quotients of affine schemes, and are therefore…

We construct small resolutions of the moduli space $\overline{Q}_n$ of stable scaled $n$-marked lines of Ziltener and Ma'u--Woodward and of the moduli space $\overline{P}_n$ of stable $n$-marked ${\mathbb G}_a$-rational trees introduced in…