Related papers: Cones of effective divisors on the blown-up P^3 in…

Let $X_0$ be a smooth projective threefold which is Fano or which has Picard number $1$. Let $\pi :X\rightarrow X_0$ be a finite composition of blowups along smooth centers. We show that for "almost all" of such $X$, if $f\in Aut(X)$ then…

We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in P^m x P^n, we describe their nef, movable and effective cones and determine when they are Mori dream spaces. Using these…

R. Beheshti showed that, for a smooth Fano hypersurface $X$ of degree $\leq 8$ over the complex number field $\mathbb{C}$, the dimension of the space of lines lying in $X$ is equal to the expected dimension. We study the space of conics on…

We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an…

We investigate the relationship between the Fano type property on fibers over a Zariski dense subset and the global Fano type property. We establish the invariance of N\'eron-Severi spaces, nef cones, effective cones, movable cones, and…

We construct a sequence of modular compactifications of the space of marked trigonal curves by allowing the branch points to coincide to a given extent. Beginning with the standard admissible cover compactification, the sequence first…

Let X be a general complex Fano threefold of genus 8. We prove that the moduli space of rank two semistable sheaves on X with Chern numbers c_1=1, c_2=6 and c_3=0 is isomorphic to the Fano surface F(X) of conics on X. Inside F(X), the…

We study Fano manifolds $X$ admitting an unsplit dominating family of rational curves and we prove that the Generalized Mukai Conjecture holds if $X$ has pseudoindex $i_X = (\dim X)/3$ or dimension $\dim X=6$. We also show that this…

In 1949 Fano published his last paper on $3$-folds with canonical sectional curves. There he constructed and described a $3$-fold of the type $X^{22}_3$ in ${\mathbb P}^{13}$ with canonical curve section, which we like to call Fano's last…

Let $(X,D)$ be log canonical pair such $\dim X = 3$ and the divisor $-(K_X + D)$ is nef and big. For a special class of such $(X,D)$'s we prove that the linear system $|-n(K_{X}+D)|$ is free for $n \gg 0$.

Let $X$ be a terminal weak $\bQ$-Fano threefold. We prove that $P_{-6}(X)>0$ and $P_{-8}(X)>1$. We also prove that the anti-canonical volume has a universal lower bound $-K_X^3 \geq 1/330$. This lower bound is optimal.

Given a smooth complex projective surface $S$ and an ample divisor $H$ on $S$, consider the blow up of $S$ along $k$ points in general position. Let $H'$ be the pullback of $H$ and $E_1,..., E_k$ be the exceptional divisors. We show that $L…

We give a proof of Mukai's Theorem on the existence of certain exceptional vector bundles on prime Fano threefolds. To our knowledge this is the first complete proof in the literature. The result is essential for Mukai's biregular…

A Mukai variety is a Fano n-fold of index n-2. In this paper we study the fundamental divisor of a Mukai variety with at worst log terminal singularities. The main result is a complete classification of log terminal Mukai varieties which…

For every $g\geq 2$ and $n\geq g+1$ we exhibit infinitely many extremal effective divisors in $\overline{\mathcal{M}}_{g,n}$ coming from the strata of abelian differentials.

For smooth projective 3-folds of general type, we prove that the relative canonical stability $\mu_s(3)\leq 8$. This is induced from our improved result of Koll\'ar: the m-canonical map of a smooth projective 3-fold of general type is…

Special covers are metacyclic covers of the projective line, with Galois group of order pm, which have a specific type of bad reduction to characteristic p. Such covers arise in the study of the arithmetic of Galois covers of the projective…

This paper is an expository survey of results about the effective divisors on moduli spaces, with a focus on what is known about the effective cones of moduli spaces of stable curves and of principally polarized abelian varieties. This…

In this paper, we check that Fano schemes of lines on certain rational cubic fourfolds are birational to Hilbert schemes of two points on K3 surfaces.

The main aim of this paper is to show that a cyclic cover of $\mathbb{P}^n$ branched along a very general divisor of degree $d$ is not stably rational provided that $n \ge 3$ and $d \ge n+1$. This generalizes the result of…