On RC varieties without smooth rational curves

Rational Curves on Varieties

Geometry of Rational Curves on Algebraic Varieties

Geometry of Rational Curves on Algebraic Varieties

Rational Curves on Minuscule Schubert Varieties

Let us denote by C the variety of lines in P3 meeting a fixed line, it is a grassmannian (and hence minuscule) Schubert variety. In [P2] we described the irreducible components of the scheme of morphisms from P1 to C and the general morphism of these irreducible components. In this text we study the scheme of morphisms from P1 to any minuscule Schubert variety X. Let us recall that we studied i...

Rational Curves on Smooth Cubic Hypersurfaces

We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in Pn of degree 2d ≤ min(n+4, 2n−2) and dimension at least three is irreducible and of the expected dimension.

Cycles on modular varieties and rational points on elliptic curves

This is a summary of a three-part lecture series given at the meeting on “Explicit methods in number theory” that was held in Oberwolfach from July 12 to 18, 2009. The theme of this lecture series was the explicit construction of algebraic points on elliptic curves from cycles on modular varieties. Given a fixed elliptic curve E over Q, the goal is to better understand the group E(Q̄) of algebra...