On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors

Abstract

We extend results on generic strange duality for K3 surfaces by showing that the proposed isomorphism holds over an entire Noether-Lefschetz divisor in the moduli space of quasipolarized K3s. We interpret the statement globally as an isomorphism of sheaves over this divisor, and also describe the global construction over the space of polarized K3s.

Strange Duality for elliptic surfaces

“…Now we have a birational isomorphism of two hyperKähler varieties, and the singular locus of the rational morphism N → M is of codimension at least two. Therefore, the singular locus of the inverse rational morphism M → N also is of codimension at least two (see [MO14], end of page 2076, for the argument).…”

“…The work on the present paper started with an attempt to strengthen the results on the Strange Duality on surfaces, and is largely motivated by the approach of Marian and Oprea [MO14]. The Strange Duality is a conjectural duality between global sections of two natural line bundles on moduli spaces of stable sheaves.…”

On a class of semihomogeneous vector bundles

“…over the moduli space of abelian surfaces endowed with a polarization of type ( 1 , 2 ), or over the moduli space of curves. The construction requires some care to kill off ambiguities; we refer the reader to [10] for details. In the curve case, the Chern characters are tautological…”

“…By varying these objects in moduli, while keeping the determinants and determinants of the Fourier–Mukai fixed to values determined by the polarization, one defines Verlinde bundles over the moduli space of abelian surfaces endowed with a polarization of type , or over the moduli space of curves. The construction requires some care to kill off ambiguities; we refer the reader to for details. In the curve case, the Chern characters are tautological In fact, explicit expressions were found and extended over the boundary, see .…”

Strange Duality on Rational Surfaces II: Higher-Rank Cases

“…Mainly there are two formulations for surfaces, one of which is due to Le Potier (see [21], [8] or §2.4 in [15]) for simply connected surfaces, while the other is due to Marian-Oprea for K3 and Abelian surfaces (see [23] or [25]). Both formulations have been studied by many people and the conjecture has been proven true for a number of cases ( [1], [2], [6], [7], [8], [15], [24], [25], [26], [28], [30], [31]). In spite of that, on strange duality for surfaces what we have known is still little.…”