Rational curves of degree 11 on a general quintic 3-fold
Twenty-five years ago Clemens conjectured that the number of smooth rational curves \(C\) of fixed degree \(d\) on a general quintic \(3\)-fold \(F\) in \(\mathbb P ^4\) is finite. Clemens' conjecture holds provided the incidence scheme NEWLINE\[NEWLINE\phi _d := \{ (C,F) \mid F\text{ is a quintic treefold in }\mathbb P ^4 \text{ containing a rational curve } C \text{ of degree }d \}NEWLINE\]NEWLINE is irreducible.NEWLINENEWLINEThe author proves in this paper that in degree \(d=11\) the incidence scheme \(\phi _{11}\) is irreducible, on a general quintic \(F\) in \(\mathbb P ^4\) there are only finitely many smooth rational curves \(C\) of degree \(11\) and each curve \(C\) is embedded in \(F\) with normal bundle \(\mathcal O (-1) \oplus \mathcal O (-1)\). Moreover the author proves that in degree \(11\) there are no singular, reduced and irreducible rational curves, nor any reduced, reducible and connected curves with rational components on \(F\). The author proved in [Commun. Algebra 33, No. 6, 1833--1872 (2005; Zbl 1079.14048)] the analogous result for \(d=10\). To do so he introduced a technique based on a combinatorial analysis of the Borel ideals that are generic initial ideals \(\mathrm{gin}(I_C)\) of the ideals \(I_C\) defining the rational curves \(C\), they are degenerations of the ideals \(I_C\). He used also the stratification of the mapping space \(\{ f: \mathbb P ^1 \rightarrow \mathbb P ^4 \mid f\text{ is a degree }d\) morphism
scientific article; zbMATH DE number 6078027