Equations of some wonderful compactifications

Hivert, Pascal

doi:10.5802/aif.2668

Cited by 1 publication

(1 citation statement)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The corresponding morphism ε : Q [2] 3 → Gr(2, V 5 ) has a rational inverse: the intersection of a line in P(V 5 ) with Q 3 is a subscheme of length 2 of Q 3 , except when the line is contained in Q 3 . The morphism ε is therefore the blow up of the scheme of lines contained in Q 3 (which is the image of the Veronese embedding v 2 : P(V 4 ) ֒→ P(Sym 2 V 4 ) = P( 2 V 5 ); see [Hi,Section 6.2]).…”

Section: The Hls Divisors D 6 and D 18mentioning

confidence: 99%

Hilbert squares of K3 surfaces and Debarre-Voisin varieties

Debarre¹,

Han²,

O’Grady³

et al. 2020

Journal De L’École Polytechnique — Mathématiques

View full text Add to dashboard Cite

The Debarre-Voisin hyperkähler fourfolds are built from alternating 3-forms on a 10-dimensional complex vector space, which we call trivectors. They are analogous to the Beauville-Donagi fourfolds associated with cubic fourfolds. In this article, we study several trivectors whose associated Debarre-Voisin variety is degenerate, in the sense that it is either reducible or has excessive dimension. We show that the Debarre-Voisin varieties specialize, along general 1-parameter degenerations to these trivectors, to varieties isomorphic or birationally isomorphic to the Hilbert square of a K3 surface. Theorem 1.3. Under a general 1-parameter deformation (σ t ) t∈∆ , the Debarre-Voisin fourfolds K σt specialize, after finite base change, to a smooth subscheme of K σ 0 which is isomorphic to S [2] , where S ⊆ X is a general K3 surface of genus 6.The limit on S [2] of the Plücker line bundles on K σt is the ample line bundle of square 22 and divisibility 2 described in Table 1. We show that it is not globally generated.1.1.4. The group G 2 × SL(3) and K3 surfaces of genus 10 (degree 18) (Section 5.2). The group G 2 is the subgroup of GL(V 7 ) leaving a general 3-form α invariant. There is a G 2 -invariant Fano 5-fold X ⊆ Gr(2, V 7 ) which has index 3 and general genus-10 K3 surfaces are obtained by intersecting X with a general 3-dimensional space10 ) are all in the same SL(V 10 )-orbit and the corresponding Debarre-Voisin variety K σ 0 splits as a product of a smooth variety of dimension 8 and of P(W ∨ 3 ) (Corollary 5.12). The excess bundle analysis shows the following (Theorem 5.15).Theorem 1.4. Under a general 1-parameter deformation (σ t ) t∈∆ , the Debarre-Voisin fourfolds K σt specialize to a smooth subscheme of K σ 0 isomorphic to S [2] , where S ⊆ X is a general genus-10 K3 surface.The limit on S [2] of the Plücker line bundles on K σt is the ample line bundle of square 22 and divisibility 2 described in Table 1. It is therefore very ample for a general genus-10 K3 surface S (Lemma 5.10).1.1.5. K3 surfaces of genus 16 (degree 30) (Section 8). This is the last case allowed by the numerical conditions of Section 3.3 where a projective model of a general K3 surface S is known. It corresponds to the last column of Table 1. Unfortunately, the current geometric knowledge for those K3 surfaces (see [Mu3]) is much weaker than in the previous cases and we were not able to find a trivector vanishing on the Hilbert square S [2] . Nevertheless, in Section 8.1, we construct on S [2] a rank-4 vector bundle with the same numerical properties as the restriction of the tautological quotient bundle of Gr(6, 10) to a Debarre-Voisin variety. In particular, this gives a rational map from S [2] to Gr(6, 10) with the expected polarization. Moduli spaces and period map2.1. Polarized hyperkähler fourfolds of degree 22 and divisibility 2 and their period map. Let X be a hyperkähler fourfold of K3 [2] -type. The abelian group H 2 (X, Z) is free abelian of rank 23 and it carries a nondegenerate integral-valued quadratic form q X (the...

show abstract

Section: The Hls Divisors D 6 and D 18mentioning

confidence: 99%