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Let M_0^R be the moduli space of smooth real cubic surfaces. We show that each of its components admits a real hyperbolic structure. More precisely, one can remove some lower-dimensional geodesic subspaces from a real hyperbolic space H^4…

We study the variation of linear sections of hypersurfaces in $\mathbb{P}^n$. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family…

Families of hyper-elliptic curves which describe the quantum moduli spaces of vacua of $N=2$ supersymmetric $SO(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the vector representation are constructed. The quantum moduli spaces…

We describe the birational geometry of the moduli space S_g^{-} of odd spin curves (theta-characteristics) for all genera g. The odd spin moduli space is a uniruled variety for g<12, and of general type for g at least 12. Furthermore, for…

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…

Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve; we study the moduli problem of parametrizing certain pairs consisting of a sheaf E on S and a map of E to a fixed reference sheaf on…

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure dimensional sheaves. Using them we establish new identifications between certain Simpson moduli…

Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected…

The strata of the moduli spaces of Abelian differentials are non-homogenous spaces carrying natural bi-algebraic structures. Partly inspired by the case of homogenous spaces carrying bi-algebraic structures (such as torii, Abelian varieties…

We show the smoothness over the affine line of the Hodge moduli space of logarithmic t-connections of coprime rank and degree on a smooth projective curve with geometrically integral fibers over an arbitrary Noetherian base. When the base…

We survey old and new results on the existence of moduli spaces of semistable coherent sheaves both in algebraic and in complex geometry.

First we give a complex ball uniformization of the moduli space of 8 ordered points on the projective line by using the theory of periods of K3 surfaces. Next we give a projective model of this moduli space by using automorphic forms on a…

Given a polytope $\sigma\subset \mathbb{R}^m$, its characteristic distribution $\delta_\sigma$ generates a $D$-module which we call the characteristic $D$-module of $\sigma$ and denote by $M_\sigma$. More generally, the characteristic…

We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.

We extend fundamental inequalities related to the canonical map of surfaces of general type to positive characteristic. Next, we classify surfaces on the Noether lines, i.e., even and odd Horikawa surfaces, in positive characteristic. We…

A 10-dimensional symplectic moduli space of torsion sheaves on the cubic 4-fold is constructed. It parametrizes the stable rank 2 vector bundles on the hypeplane sections of the cubic 4-fold which are obtained by Serre's construction from…

Mukai proved that the moduli space of simple sheaves on a smooth projective K3 surface is symplectic, and in \cite{FM2} we gave two constructions allowing one to construct new locally closed Lagrangian/isotropic subspaces of the moduli from…

For any almost-simple group $G$ over an algebraically closed field $k$ of characteristic zero, we describe the automorphism group of the moduli space of semistable $G$-bundles over a connected smooth projective curve $C$ of genus at least…

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…

We study the locus of smooth hypersurfaces inside the Hilbert scheme of a smooth projective complex variety. In the spirit of scanning, we construct a map to a continuous section space of a projective bundle, and show that it induces an…