Lagrangian subbundles and codimension 3 subcanonical subschemes

Abstract

We show that a Gorenstein subcanonical codimension 3 subscheme Z ⊂ X = P N , N ≥ 4, can be realized as the locus along which two Lagrangian subbundles of a twisted orthogonal bundle meet degenerately, and conversely. We extend this result to singular Z and all quasiprojective ambient schemes X under the necessary hypothesis that Z is strongly subcanonical in a sense defined below. A central point is that a pair of Lagrangian subbundles can be transformed locally into an alternating map. In the local case our structure theorem reduces to that of Buchsbaum-Eisenbud [6] and says that Z is Pfaffian.We also prove codimension one symmetric and skew-symmetric analogues of our structure theorems.

Construction of Calabi-Yau $3$-folds in $\mathbb P^6$

“…A recent result of Walter [Wal96] shows that under a mild additional hypothesis every subcanonical Gorenstein codimension 3 subscheme X in P n is Pfaffian (see [EPW01] for a description of the non-Pfaffian case), and therefore one can attempt to get its equations, starting by constructing its Pfaffian resolution.In this paper we apply this method to build examples of smooth Calabi-Yau 3-folds in P 6 . In this case we show the existence of three unirational components of their Hilbert scheme, all having the same dimension 23 + 48 = 71.…”

“…A recent result of Walter [Wal96] shows that under a mild additional hypothesis every subcanonical Gorenstein codimension 3 subscheme X in P n is Pfaffian (see [EPW01] for a description of the non-Pfaffian case), and therefore one can attempt to get its equations, starting by constructing its Pfaffian resolution. Being Pfaffian, this subscheme is automatically subcanonical, in the sense that its canonical bundle is the restriction of a multiple of O P n (1).…”

On the Chow groups of certain EPW sextics

“…It is known that a variety X as in theorem 3.1 has a birational model X ′ which is a hyperkähler variety; X ′ is a so-called double EPW sextic [30], [29], [32]. The variety X ′ has a generically 2 : 1 morphism to a slightly singular sextic hypersurface Y ⊂ P 5 , called an EPW sextic [10], [30]. Theorem 3.1 has interesting consequences for the Chow ring of this EPW sextic:…”

Hyper-Kähler fourfolds and Kummer surfaces

“…Section 3 is devoted to the construction of elements of as double covers of appropriate Lagrangian degeneracy loci inside a cone over the Segre embedding of . This construction is analogous to the construction of Eisenbud‐Popescu‐Walter sextics called EPW sextics . It is also naturally related to special EPW cubes .…”

“…In Section 2, we describe a ‘baby case’ of our constructions by presenting two constructions of the Kummer surfaces first as Lagrangian degeneracy loci (as in [, Theorem 9.2]) and next as a quotient of the base of a fibration on the Hilbert scheme of ‐conics on a quadric section of a cone over the Segre embedding . The relation to the description of the EPW quartic section is explained in Section 4.1.…”