Coisotropic deformations of algebraic varieties and integrable systems

Coisotropic deformations of algebraic varieties are defined as those for which an ideal of the deformed variety is a Poisson ideal. It is shown that coisotropic deformations of sets of intersection points of plane quadrics, cubics and space algebraic curves are governed, in particular, by the dKP, WDVV, dVN, d2DTL equations and other integrable hydrodynamical type systems. Particular attention is paid to the study of two- and three-dimensional deformations of elliptic curves. Problem of an appropriate choice of Poisson structure is discussed.Comment: 17 pages, no figure

The Witten-Kontsevich Theorem

openThe Witten-Kontsevich theorem relates intersection products of certain cohomology classes in the tautological ring of the moduli space of stable curves, to the KdV hierarchy of partial differential equations. In this thesis, a recent proof of this theorem is presented. Firstly, the ELSV formula relates such intersection products to simple Hurwitz numbers, which count branched covers of algebraic curves. Subsequently, the link between Hurwitz theory and integrable systems is made via the Sato Grassmannian construction for the KP hierarchy.The Witten-Kontsevich theorem relates intersection products of certain cohomology classes in the tautological ring of the moduli space of stable curves, to the KdV hierarchy of partial differential equations. In this thesis, a recent proof of this theorem is presented. Firstly, the ELSV formula relates such intersection products to simple Hurwitz numbers, which count branched covers of algebraic curves. Subsequently, the link between Hurwitz theory and integrable systems is made via the Sato Grassmannian construction for the KP hierarchy

Intersection Number of Plane Curves

In algebraic geometry, seemingly geometric problems can be solved using algebraic techniques. Some of the most basic geometric objects we can study are polynomial curves in the plane. In this paper we focus on the intersections of two curves. We address both the number of times two curves intersect at a given point, counting multiplicity (whatever that means), and the total number of intersections of the curves, again counting multiplicity. The former is known as the intersection number of the curves at the point. This concept, although geometrically motivated, can be described in algebraic terms; it is this relationship which makes it such a powerful concept. The paper concludes with an important application of the intersection number, Bezout\u27s Theorem. This ubiquitous theorem gives a beautifully concise solution to the total number of intersections, given sufficiently nice assumptions on the curves and the ambient space

Intersection Number of Plane Curves

In algebraic geometry, seemingly geometric problems can be solved using algebraic techniques. Some of the most basic geometric objects we can study are polynomial curves in the plane. In this paper we focus on the intersections of two curves. We address both the number of times two curves intersect at a given point, counting multiplicity (whatever that means), and the total number of intersections of the curves, again counting multiplicity. The former is known as the intersection number of the curves at the point. This concept, although geometrically motivated, can be described in algebraic terms; it is this relationship which makes it such a powerful concept. The paper concludes with an important application of the intersection number, Bezout\u27s Theorem. This ubiquitous theorem gives a beautifully concise solution to the total number of intersections, given sufficiently nice assumptions on the curves and the ambient space