Nested varieties of K3 type

Abstract

Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0.

Fano 3-folds from homogeneous vector bundles over Grassmannians

“…The two extra sections of O(0, 1) cut P 3 in a codimension two linear subspace. Therefore our zero locus can be seen as the blow up of 3,5). By Lemma 2.5 this can be easily identified with the alternative description of this Fano given in [9].…”

“…The ampleness of U| Y ⊗ L required in Lemma 2.9 is not necessary, as shown in [3,Proposition 46]. We remark that…”

“…Our choice fell on [16], for which an already developed package [17] implementing the methods we need is available. This allows us to compute (−K Z (F ) ) 3 , as the Chern classes of the canonical sheaf of Z (F) are easy to express. As for h 0 (−K Z (F ) ), we certainly know how to compute χ(−K Z (F ) ).…”

“…For j = 1, (2) yields the usual conormal short exact sequence. The term on the left is F ∨ | Y , which is resolved by a Koszul complex whose terms are The middle term is X | Y , which is resolved by a Koszul complex whose terms are 3…”

“…In [9] this variety is described as Z (O(1, 0)⊕O(0, 1) ⊕3 ) ⊂ Fl(1, 2, 5). This description is equivalent to the one given here simply applying Lemma 2.2 with k = 3 (where we identify Gr (3,4) and Gr(3, 5) with P 3 and Gr (2,5)). The three residual sections of O(0, 1) cut both Gr(2, 5) (in V 5 ) and Gr(2, 4) (in a conic).…”