A Note on Rational Curves on General Fano Hypersurfaces

Abstract

We show the Kontsevich space of rational curves of degree at most roughly 2− √ 2 2 n on a general hypersurface X ⊂ P n of degree n − 1 is equidimensional of expected dimension and has two components: one consisting generically of smooth, embedded rational curves and the other consisting of multiple covers of a line. This proves more cases of a conjecture of Coskun, Harris, and Starr and shows the Gromov-Witten invariants in these cases are enumerative.

Low-degree Rational curves on hypersurfaces in projective spaces and their degenerations

“…For d ≤ n − 2 and e arbitrary, this has been proven by Riedl-Yang [10] based on bend-andbreak. For d = n − 1 some results for low e were obtained by Tseng [11].…”

“…are both admit balanced decompositions. We will say that a line subbundle of E 12 is of type (11) if its restrictions on C * 1 and C 2 are both of the upper degree, and likewise for (10) etc, likewise for (111) etc. Assume to begin with that r + (E * 1 ) + r + (E 2 ) > r. Then by induction, E 12 is a sum of line subbundles of types (11), (01) and (10), where the first two make up the upper subspace at p 2 .…”

“…We will say that a line subbundle of E 12 is of type (11) if its restrictions on C * 1 and C 2 are both of the upper degree, and likewise for (10) etc, likewise for (111) etc. Assume to begin with that r + (E * 1 ) + r + (E 2 ) > r. Then by induction, E 12 is a sum of line subbundles of types (11), (01) and (10), where the first two make up the upper subspace at p 2 . By (2) (together with general position), the first two types may be glued to a direct summand line subbundle of type 0 on E 3 , while the last type may be glued to summands to type 0 or 1.…”

“…This conjecture has been proven for d ≤ n − 2 by Riedl and Yang [10] which also contains extensive references. See also [11] for a partial extension to the case d = n − 1.…”