Normal bundles of rational curves on complete intersections

Abstract

Let X ⊂ P n be a general Fano complete intersection of type (d 1 , . . . , d k ). If at least one d i is greater than 2, we show that X contains rational curves of degree e ≤ n with balanced normal bundle. If all d i are 2 and n ≥ 2k + 1, we show that X contains rational curves of degree e ≤ n − 1 with balanced normal bundle. As an application, we prove a stronger version of the theorem of Z. Tian [Ti15], Q. Chen and Y. Zhu [CZ14] that X is separably rationally connected by exhibiting very free rational curves in X of optimal degrees.

Koszul modules and Green’s conjecture

“…We write ι ∨ : 2 V ∨ ։ K ∨ for the dual to the inclusion ι, and let K ⊥ := Ker(ι ∨ ) ⊆ 2 V ∨ . It is shown in [24,Lemma 2.4] that the set-theoretic support of W (V, K) in the affine space V ∨ is given by the resonance variety R(V, K), defined as (5) R(V, K) := a ∈ V ∨ : there exists b ∈ V ∨ such that a ∧ b ∈ K ⊥ \ {0} ∪ {0}.…”

“…In particular, W (V, K) has finite length if and only R(V, K) = {0}. In view of (5), this last condition is equivalent to the fact that the linear subspace PK ⊥ ⊆ P( 2 V ∨ ) is disjoint from the Grassmann variety…”

Universal degeneracy classes for vector bundles on $\mathbb{P}^1$ bundles

“…The splitting of this bundle has been studied extensively for X = P n (see for example [1,4,10,11,24,25,26]) and for other varieties (e.g. [6,5,16,20,19]).…”