Quantum cohomology of \normalfont𝐻𝑖𝑙𝑏²(ℙ¹×ℙ¹) and enumerative applications

Pontoni, Dalide

doi:10.1090/S0002-9947-07-04169-4

Quantum cohomology of $\text {\normalfont Hilb}^2(\mathbb {P}^1 \times \mathbb {P}^1)$ and enumerative applications
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Trans. Amer. Math. Soc. 359 (2007), 5419-5448 Request permission

Abstract:

We compute the small quantum cohomology of $\mathbf {H}=$Hilb$^2(\mathbb {P}^1\times \mathbb {P}^1)$ and determine recursively most of the big quantum cohomology. We prove a relationship between the invariants so obtained and the enumerative geometry of hyperelliptic curves in $\mathbb {P}^1\times \mathbb {P}^1$. This extends the results obtained by Graber (2001) for Hilb$^2(\mathbb {P}^2)$ and hyperelliptic curves in $\mathbb {P}^2$.

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