Degeneracy loci of morphisms between vector bundles have been used in a wide variety of situations. We introduce a vast generalization of this notion, based on orbit closures of algebraic groups in their linear representations. A preferred class of our orbital degeneracy loci is characterized by a certain crepancy condition on the orbit closure, that allows to get some control on the canonical sheaf. This condition is fulfilled for Richardson nilpotent orbits, and also for partially decomposable skew-symmetric three-forms in six variables. In order to illustrate the efficiency and flexibility of our methods, we construct in both situations many Calabi-Yau manifolds of dimension three and four, as well as a few Fano varieties, including some new Fano fourfolds.
In [BFMT17] we introduced orbital degeneracy loci as generalizations of degeneracy loci of morphisms between vector bundles. Orbital degeneracy loci can be constructed from any stable subvariety of a representation of an algebraic group. In this paper we show that their canonical bundles can be conveniently controlled in the case where the affine coordinate ring of the subvariety is Gorenstein. We then study in a systematic way the subvarieties obtained as orbit closures in representations with finitely many orbits, and we determine the canonical bundles of the corresponding orbital degeneracy loci in the Gorenstein cases. Applications are given to the construction of low dimensional varieties with negative or trivial canonical bundle.
We rework the Mori–Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.
We prove that, for 3 < m < n − 1, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension n is birational to the Hilbert scheme of the degeneracy loci of m global sections of Ω P n−1 (2), the twisted cotangent bundle on P n−1 . For 3 = m < n − 1 and n odd, this Grassmannian is proved to be birational to the set of Veronese surfaces parametrized by the Pfaffians of linear skew-symmetric matrices of order n.ON THE HILBERT SCHEME OF DEGENERACY LOCI OF O m P(V ) → Ω P(V ) (2) 2 the Palatini scrolls in P 5 : the main result of [FM02] states that ρ is birational when (m, n) = (4, 6).In the case (m, n) = (3, 6), however, it was proved in [BM01], and in fact classically known to Fano [Fan30], that ρ is dominant and generically 4 : 1. Other cases have been recently studied in [FF10b]. Our main result is a complete description of the features of the map ρ.Theorem. Let m, n ∈ N satisfying 2 < m < n − 1 and letbe the rational morphism introduced in (1), sending the class of a morphism φ : O m P(V ) → Ω P(V ) (2) to its degeneracy locus X φ , considered as a point in the Hilbert scheme.i. If m ≥ 4 or (m, n) = (3, 5), then ρ is birational; in particular, the Hilbert scheme H is irreducible and generically smooth of dimension m n 2 − m . ii. If m = 3 and n = 6, then ρ is generically injective. Moreoverii.a. if n is odd, ρ is dominant on a closed subscheme H ′ of H of codimension n 8 (n − 3)(n − 5). The general element of H is a general projection in P(V ) of a Veronese surface v n−1 2 (P 2 ), embedded via the complete linear system of curves of degree n−1 2 ; in particular, H is irreducible. The general element of H ′ is a particular projection in P(V ), obtained using the linear space spanned by the partial derivatives of order n−52 ). Part i. of the Theorem is the content of Theorem 20 and Corollary 21; the general injectivity of ρ will be proved in Theorem 15. In the case m = 3, the codimensions of H ′ in H are computed in Proposition 22; if n is odd, the characterization of the general element of H ′ is performed in Theorem 23, while the general element of H is described in Proposition 29. In the case n even, this was done in [FF10b].This theorem provides a complete description, showing that the case (m, n) = (3, 6) is the unique in which ρ is not generically injective. It shows also that, for m = 3, the case n = 5 is the only one in which we have birationality. The missing birationality for an odd n > 6 can be explained by means of the above description of Im(ρ) ⊂ H: the general projection of a Veronese surface is not special in the sense of the Theorem, so it is not in the image of ρ. For small values of m, n, this theorem provides another proof of the classical results already known; it also covers the main results of [FF10b].The main tool for performing the cohomology computations needed to prove the Theorem is the so-called Kempf-Lascoux-Weyman's method of calculation of syzygies via resolution of singularities; the original idea of Kempf was that the direct ima...
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