#### 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).

Done