Related papers: Minimal rational curves on complete symmetric vari…

Esnault asked whether every smooth complex projective variety with infinite fundamental group has a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). In a sense, this would mean that every variety with…

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral…

Recently, Kanemitsu has discovered a counterexample to the long-standing conjecture that the tangent bundle of a Fano manifold of Picard number one is (semi)stable. His counterexample is a smooth horospherical variety. There is a weaker…

The goal of this work is to study geometric properties of geometrically irreducible subschemes on degenerations of Fano varieties (more generally, of separably rationally connected varieties). It is known that these geometrically…

The symmetric projective varieties of rank one are all smooth and Fano by a classic result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric $G$-varieties of rank 2 which are Fano. When $G$ is…

We investigate basic properties of uniformly rational varieties, i.e. those smooth varieties for which every point has a Zariski open neighborhood isomorphic to an open subset of A^n. It is an open question of Gromov whether all smooth…

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three - with one…

A complex projective manifold is rationally connected, resp. rationally simply connected, if finite subsets are connected by a rational curve, resp. the spaces parameterizing these connecting rational curves are themselves rationally…

We give lower bounds for the degree of the discriminant with respect to y of separable polynomials f in K[x,y] over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a…

We study symplectic geometry of rationally connected $3$-folds. The first result shows that rationally connectedness is a symplectic deformation invariant in dimension $3$. If a rationally connected $3$-fold $X$ is Fano or $b_2(X)=2$, we…

We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with…

As we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which are in one-to-one correspondence with singular…

Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.

We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semi-simple group and H is the fixed subgroup for an involutorial automorphism of G. We find generators…

A rotation-minimizing frame $({\bf f}_1,{\bf f}_2,{\bf f}_3)$ on a space curve ${\bf r}(\xi)$ defines an orthonormal basis for $\mathbb{R}^3$ in which ${\bf f}_1={\bf r}'/|{\bf r}'|$ is the curve tangent, and the normal-plane vectors ${\bf…

We investigate families of minimal rational curves on Schubert varieties, their Bott-Samelson desingularizations, and their generalizations constructed by Nicolas Perrin in the minuscule case. In particular, we describe the minimal families…

We present bounds for the geometric degree of the tangent bundle and the tangential variety of a smooth affine algebraic variety $V$ in terms of the geometric degree of $V$. We first analyze the case of curves, showing an explicit relation…

Complex contact manifolds arise naturally in differential geometry, algebraic geometry and exterior differential systems. Their classification would answer an important question about holonomy groups. The geometry of such manifold $X$ is…

In this paper, we construct some families of infinitely many hyperelliptic curves of genus $2$ with exactly two rational points. In the proof, we first show that the Mordell-Weil ranks of these hyperelliptic curves are $0$ and then…

We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index e, then the degree of irrationality of a very general complex Fano hypersurface of index e and dimension n…