Normal bundles of rational curves on complete intersections

Abstract

Let [Formula: see text] be a general Fano complete intersection of type [Formula: see text]. If at least one [Formula: see text] is greater than [Formula: see text], we show that [Formula: see text] contains rational curves of degree [Formula: see text] with balanced normal bundle. If all [Formula: see text] are [Formula: see text] and [Formula: see text], we show that [Formula: see text] contains rational curves of degree [Formula: see text] with balanced normal bundle. As an application, we prove a stronger version of the theorem of Tian [27], Chen and Zhu [4] that [Formula: see text] is separably rationally connected by exhibiting very free rational curves in [Formula: see text] of optimal degrees.

Koszul modules and Green’s conjecture

“…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…”

“…In Theorem 5.2 we also give a similar characterization for the (non-)vanishing of b i,2 when 3 ≤ p ≤ g+1 2 , and in Section 5.9 we consider p = 2. To prove Theorem 5.2, we realize the groups K i,2 T , O T (1) as graded components of Weyman modules as explained next, and then apply Theorem 1.3 and the explicit description (5) of the set-theoretic support of a Koszul module.…”

Very free rational curves in Fano varieties

“…Given a rational curve C on X, the normal bundle N C|X controls the deformations of C in X and carries essential information about the local structure of the space of rational curves. Consequently, the normal bundles of rational curves have been studied extensively when X is P n ([AR17, Con06, CR18, EV81, EV82, GS80, Ran07, Sa82, Sa80]) and more generally (see for example [Br13,CR19,K96,LT19,Sh12b]).…”

Rational curves on del Pezzo surfaces in positive characteristic

“…Let us focus on results in positive characteristic. First of all, there are many papers which study the separable rational connectedness of smooth Fano varieties in characteristic p (for example [She10], [Zhu11], [CZ14], [GLP + 15], [Tia15], [CR19], [ST19], and [CS21]). János Kollár asked whether any smooth Fano variety is separably rationally connected, but this question is wide open at this moment.…”