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We use Gromov-Witten theory to study rational curves in holomorphic symplectic varieties. We present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general…

Let C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map…

We pose some questions about spaces parametrizing rational curves on rationally connected varieties. We give a partial answer for cubic threefolds. Many of our results were previously proved by Iliev, Markushevich and Tikhimirov by…

Let $X$ be a smooth quasi-projective algebraic surface and let $\Delta_n$ the big diagonal in the product variety $X^n$. We study cohomological properties of the ideal sheaves $\mathcal{I}^k_{\Delta_n}$ and their invariants…

We construct inductively an equivariant compactification of the algebraic group ${\mathbb W}_n$ of Witt vectors of finite length over a field of characteristic $p>0$. We obtain smooth projective rational varieties $\bar{\mathbb W}_n$,…

In the open problem of classification of rational cuspidal plane curves it is essential to find good necessary conditions on the type of singularities of a curve C in order C to exit. Motivated by the study of the Seiberg-Witten invariant…

We discuss the space of sections and certain bisections on a quadric surfaces bundle $X$ over a smooth curve. The Abel-Jacobi from these spaces to the intermediate Jacobian will be shown to be dominant with rationally connected fibers. As…

Let $X \subset \mathbb{P}(w_0, w_1, w_2, w_3)$ be a quasismooth well-formed weighted projective hypersurface and let $L = lcm(w_0,w_1,w_2,w_3)$. We characterize when $X$ is rational under the assumption that $L$ divides $deg(X)$ by…

We study families of rational curves on certain irreducible holomorphic symplectic varieties. In particular, we prove that any ample linear system on a projective holomorphic symplectic variety of K3[n]-type contains a uniruled divisor. As…

Prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. We define and construct explicitly prescribed rational functions-valued bundle $\mathcal{W}_M$ over a smooth…

In this paper we study smooth complex projective polarized varieties (X,H) of dimension n \ge 2 which admit a dominating family V of rational curves of H-degree 3, such that two general points of X may be joined by a curve parametrized by…

We give a criterion for a continuous family of curves on a nodal $K$-trivial threefold $X_0$ to contribute geometrically rigid curves to a general smoothing of $X_0$. As an application, we prove the existence of geometrically rigid curves…

To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed…

We study a double solid X branched along a nodal sextic surface in a projective space and the 2-torsion subgroup in the third integer cohomology group of a resolution of singularities of X. This group can be considered as an obstruction to…

We generalize the notion of tight geodesics in the curve complex to tight trees. We then use tight trees to construct model geometries for certain surface bundles over graphs. This extends some aspects of the combinatorial model for doubly…

We show that the Burkhardt quartic threefold is rational over any field of characteristic distinct from 3. We compute its zeta function over finite fields. We realize one of its moduli interpretations explicitly by determining a model for…

This article extends the study of cyclic ramified covers of the projective line defined by Kummer equations. We consider the most general case of such covers, allowing arbitrary orders in the roots of the generating radicant. The primary…

We determine all triples $(e,d,n)$ for which a general degree $d$ hypersurface $X\subset \mathbb{P}^n$ contains a degree $e$ rational curve $C$ with balanced restricted tangent bundle $T_X|_C$. In addition, we show how to compute explicit…

We develop an intersection theory for a singular hemitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety. And we prove the existence of a natural rational fibration structure…

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to…