Universal Series for Hilbert Schemes and Strange Duality

Abstract

We examine a sequence of examples of pairs of moduli spaces of sheaves on P 2 where Le Potier's strange duality is expected to hold. One of the moduli spaces in these pairs is the Hilbert scheme of two points. We compute the sections of the relevant theta bundle as a representation of SL(X), where P 2 = P(X). For the higher rank space, we construct a moduli space using the resolution of exceptional bundles from Coskun, Huizenga, and Woolf [CHW14]. We compute a subspace of the sections of the theta bundle which is dual to the sections on the Hilbert scheme.In the second part, we use a result from Goller and Lin [GL19] to rigorously apply the "finte Quot scheme method", introduced in Marian and Oprea [MO07] and Bertram, Goller, and Johnson [BGJ16]. This requires us to prove that the kernels in appearing in the Quot scheme have the appropriate resolution by exceptional bundles.

The combinatorics of Lehn's conjecture

“…In the surface case, strange duality was pursued along these lines in [2]. We sharpen the conjectures in [5], providing closed formulas for some of the series involved. Specifically, consider a pair (S, V ) where V is a rank s vector bundle on a nonsingular projective surface S. The associated vector bundle V [n] on the Hilbert scheme has rank sn.…”

“…For surfaces, a complete higher rank analogue of Lehn's conjecture is an open question. In this direction, several conjectures were recently formulated by D. Johnson [5], relating Segre theory to Verlinde theory in the Hilbert scheme context. Johnson's formulation of the conjectures was inspired by counts of points of 0-dimensional Quot schemes and strange duality, similar to the strategy used to prove strange duality for curves in [8].…”

“…Here, H n denotes the line bundle induced by H = det V on the symmetric product, and E is − 1 2 of the exceptional divisor. By [5] and [4], the two series corresponding to K-trivial surfaces are determined in closed form f r (z) = (1 + t) r 2 1 + r 2 t , g r (z) = 1 + t after the change of variables…”

“…For rank s = 1 (corresponding to r = 0), the Chern class c2n(V [n] ) is trivial for n > 0 since V [n] is only of rank n. The formulas of Conjecture 1 are singular in the r = 0 case 5. We have a0 = a±1 = b0 = b±1 = 1 .…”

The geometry of Hilbert schemes of two points on projective space

“…When F is a line bundle, these have been of extensive interest, e.g. [Leh,MOP1,Voi,MOP2,Joh]. As pr 2 is a finite flat morphism of degree 2, F [2] is a rank 2 vector bundle in that case.…”