Spaces of Low-Degree Rational Curves on Fano Complete Intersections

Abstract

For a smooth complete intersection X, we consider a general fiber F of the following evaluation map ev of the Kontsevich moduli space ev : M 0,m (X, m) → X m and the forgetful map F : F → M 0,m . We prove that a general fiber of the map F is a smooth complete intersection if X is of low degree. The result sheds some light on the arithmetic and geometry of Fano complete intersections.Date: November 12, 2018.1 This condition is to guarantee that the locus of Ft parametrizing the stable maps of maximal degeneration type is of dimension at least 2, cf. Corollary 5.7 2 The map Φ is introduced in the proof of [dJS06, Lemma 6.4]. 3 We denote by P(V )/P(W ) the projective space P(V /W ) for a flag (W ⊆ V ) of a vector space V and denote by Span(D) the smallest projective subspaces containing the algebraic set D(⊆ P n ). 4 The fiber Ft is possibly empty. However, in Section 5, we will see the conditions (1.2) andLemma 2.8 guarantee that Ft is a non-empty, cf. Proposition 5.6. 5 Strong approximation is considered to be more difficult to show than weak approximation.

Strong approximation over function fields

“…Identifying the boundary ∆ as a complete intersection. Fixing a general (p, q), the geometry of the pair (F ′ , ∆ ′ ) has been studied in [Pan13]. Let us recall the basic construction.…”

Lines, conics, and all that

“…In [422] one can find restrictions under which the Kontsevich moduli space M 0,0 (X, e) is of general type. See also [361] for studies on the Kontsevich moduli spaces of rational curves through marked points in a Fano complete intersection variety.…”