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State of the Art Seminar

Wednesday, November 22, 2017 - 15:30
Seminar Hall
Anantadulal Paul
Counting complex curves with singularity in CP^2

The study of of degree d curves in $\mathbb{CP}^2$ is one of the interesting topic which arises in the context of Enumerative Geometry. The specific problem, which I will talk about, can be stated as follows:

Let $\mathbb{CP}^2$ be a compact complex surface and $L \rightarrow \mathbb{CP}^2$ a holomorphic line bundle that is sufficiently ample. Let $\mathcal{D} := \mathbb{P}^\frac{d(d+3)}{2}$ be the space of all degree d complex curves in $\mathbb{CP}^2$ - What is $\mathcal{N}(\mathfrak{X}_{k})$, the number of curves in $\mathbb{CP}^2$, passing through $\frac{d(d+3)}{2} -k$ generic points, having singularity of type $\mathfrak{X}_{k}$, where $k$ is the codimension of the singularity $\mathfrak{X}_{k}$?

In this talk we will describe what is a zero set of vector bundle over $\mathbb{CP}^2$ and then calculate $\mathcal{N}({A_1})$ for degree d complex curves in $\mathbb{CP}^2$. Here
$\mathcal{N}({A_1})$ represents number of degree d curves in $\mathbb{CP}^2$ passing through $\frac{d(d+3)}{2} -1$ points having $A_1$ singularity. 

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