Projective Structures, Neighborhoods of Rational Curves and Painlevé equations

Painlevé VI Equations with Algebraic Solutions and Family of Curves

In families of Painlevé VI differential equations having common algebraic solutions we classify all the members which come from geometry, i.e. the corresponding linear differential equations which are Picard-Fuchs associated to families of algebraic varieties. In our case, we have one family with zero dimensional fibers and all others are families of curves. We use the classification of familie...

Painlevé equations and deformations of rational surfaces with rational double points ∗ †

In this paper we give an answer to the fundamental questions about the Painlevé equations. Where do the Bäcklund transformations come from? Our approach depends on the geometry of the projective surface constructed by Okamoto and reviewed in [U2]. The Painlevé equations were discovered around 1900 in the pursuit of special functions. Painlevé and Gambier classified algebraic differential equati...

Rational Curves and Ordinary Differential Equations

Complex analytic 2nd order ODE systems whose solutions close up to become rational curves are characterized by the vanishing of an explicit differential invariant, and form an infinite dimensional family of integrable systems.

Counting Rational Curves of Arbitrary Shape in Projective Spaces

Counting rational curves of arbitrary shape in projective spaces

We present an approach to a large class of enumerative problems concerning rational curves in projective spaces. This approach uses analysis to obtain topological information about moduli spaces of stable maps. We demonstrate it by enumerating one-component rational curves with a triple point or a tacnodal point in the three-dimensional projective space and with a cusp in any projective space. ...