Related papers: On rational curves in n-space with given normal bu…

Let G/Q be an homogeneous variety embedded in a projective space P thanks to an ample line bundle L. Take a projective space containing P and form the cone X over G/Q, we call this a cone over an homogeneous variety. Let $\alpha$ a class of…

For any standard quadric surface bundle over $\mathbb P^2$, we show that the locus of rational fibres is dense in the moduli space.

Let M be the moduli space of stable bundles of rank 2 and with fixed determinant \mathcal{L} of degree d on a smooth projective curve C of genus g>= 2. When g=3 and d is even, we prove, for any point [W]\in M, there is a minimal rational…

We prove that moduli spaces of torsion-free sheaves on a projective smooth complex surface are irreducible, reduced and of the expected dimension, provided the expected dimension is large enough. Actually we prove more: given a line bundle…

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of arithmetic genus 3 contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the…

Let k be an algebraically closed field of characteristic zero. Let f:X-->S be a flat, projective morphism of k-schemes of finite type with integral geometric fibers. We prove existence of a projective relative moduli space for semistable…

We show that each of the irreducible components of moduli of rank 2 torsion-free sheaves with odd Euler characteristic over a reducible nodal curve is rational.

We study the moduli space of stable sheaves of Euler characteristic 1 supported on curves of bidegree (3, 3) contained in a smooth quadric surface. We show that this moduli space is rational. We compute its Betti numbers by studying the…

The moduli spaces of trigonal curves of odd genus $g>4$ are proven to be rational.

We give a proof of the Gromov compactness theorem using the language of stable curves (i.e. cusp-curve of Gromov, or stable maps of Kontsevich and Manin) in general setting: An almost complex structure on a target manifold is only…

We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.

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…

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…

Let $X$ be a smooth projective Fano variety over the complex numbers. We study the moduli spaces of rational curves on $X$ using the perspective of Manin's Conjecture. In particular, we bound the dimension and number of components of spaces…

We study the moduli problem of pairs consisting of a rank 2 vector bundle and a nonzero section over a fixed smooth curve. The stability condition involves a parameter; as it varies, we show that the moduli space undergoes a sequence of…

We show that $\mathcal{M}_{g,n}$, the moduli space of smooth curves of genus $g$ together with $n$ marked points, is unirational for $g=12$ and $2 \leq n\leq 4$ and for $g=13$ and $1 \leq n \leq 3$, by constructing suitable dominant…

We describe the irreducible components of the moduli spaces of rational curves on Artin-Mumford double solids. This provides the first example of Fano varieties that satisfy Geometric Manin's Conjecture with multiple Manin components in…

Given an automorphism of a smooth complex algebraic curve, there is an induced action on the moduli space of semi-stable rank 2 holomorphic bundles with fixed determinant. We give a complete description of the fixed variety in terms of…

We show that the moduli space of stable n-pointed rational curves $\overline{M}_{0,n}$ with its boundary $\Delta$ is algebraically hyperbolic.

We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker-Maruyama moduli space $M(e,n)$ of rank 2 stable vector bundles with the first Chern class $e=0$ or -1 and all possible…