The virtual K-theory of Quot schemes of surfaces

Abstract We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K -theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all surfaces and dimension 1 quotients for surfaces X with p g > 0 . We also show that the generating series of virtual cobordism classes can be irrational. Given a K -theory class on X of rank r , we associate natural series of virtual Segre and Verlinde numbers. We show that the Segre and Verlinde series match in the following cases: • [(i)] Quot schemes of dimension 0 quotients, • [(ii)] Hilbert schemes of points and curves over surfaces with p g > 0 , • [(iii)] Quot schemes of minimal elliptic surfaces for quotients supported on fiber classes. Moreover, for punctual quotients of the trivial sheaf of rank N , we prove a new symmetry of the Segre/Verlinde series exchanging r and N . The Segre/Verlinde statements have analogues for punctual Quot schemes over curves.

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