Finite Quot schemes and strange duality

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Assume a general sheaf in M(e)M(e) is locally free, let VV have Chern character e∗+fe^*+f, and assume χ(e⋅f)=0\chi(e\cdot f)=0. Let Quot⁡(V,f)\operatorname{Quot}(V,f) parametrize quotients V↠FV\twoheadrightarrow F with ch⁡(F)=f\operatorname{ch}(F)=f. In the setting above, suppose Quot⁡(V,f)\operatorname{Quot}(V,f) is finite and reduced. Finite Quot scheme strange-duality conjecture.

#Quot⁡(V,f)=h0(M(e),OM(e)(Θf))=h0(M(f),OM(f)(Θe)).\#\operatorname{Quot}(V,f)=h^0(M(e),\mathcal{O}_{M(e)}(\Theta_f))=h^0(M(f),\mathcal{O}_{M(f)}(\Theta_e)).

This is a numerical formulation of strange duality in the finite, reduced Quot-scheme setting. The source presents the assertion as a conjectural reduction and does not establish it in general.

References

Primary source

Thomas Goller and Yinbang Lin, “Rank-one sheaves and stable pairs on surfaces”, arXiv:1907.05180 (2019).

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