Alladi Ramakrishnan Hall

Enumerative Geometry of rational cuspidal curves on del-Pezzo surfaces

Ritwik Mukherjee

TIFR, Mumbai

Enumerative geometry is a branch of mathematics that deals 
with the following question: "How many geometric objects are there that 
satisfy certain constraints?" The simplest example of such a question is 
"How many lines pass through two points?". A more interesting question 
is "How many lines are there in three dimensional space that intersect 
four generic lines?". An extremely important class of enumerative question 
is to ask "How many rational (genus 0) degree d curves are there in 
CP^2 that pass through 3d-1 generic points?" Although this question 
was investigated in the nineteenth century, a complete solution to this 
problem was unknown until the early 90's, when Kontsevich-Manin 
and Ruan-Tian announced a formula. In this talk we will discuss some 
natural generalizations of the above question; in particular we will be looking 
at rational curves on del-Pezzo surfaces that have a cuspidal singularity. We 
will describe a topological method to approach such questions. If time 
permits, we will also explain the idea of how to enumerate genus one 
curves with a fixed complex structure by comparing it with the Symplectic 
Invariant of a manifold (which are essentially the number of curves that are 
solutions to the perturbed d-bar equation).