Deformations of a morphism along a foliation and applications.

*(English)*Zbl 0659.14008
Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, part 1, Proc. Symp. Pure Math. 46, No. 1, 245-268 (1987).

[For the entire collection see Zbl 0626.00011.]

The author studies algebraic smooth foliations on smooth schemes X. They are by definition the involutive subbundles of the tangent sheaf \(T_ X\), which are integrable. The integrability is defined in a usual manner when the base scheme contains \({\mathbb{Q}}\) while, if the base field is of positive characteristic, then it is defined along with a lift to characteristic 0. The author studies a holomorphic map \(f: Y\to X,\) with a smooth foliation \({\mathcal F}\) on X (proper smooth, over characteristic 0), and considers deformations of f along the foliation, i.e. those deformations whose infinitesimal deformations are in the pull back of \({\mathcal F}\). The existence of universal formal deformations is proved along the line of Grothendieck’s SGA (Sémin. géométrie algébrique). As an application, it is proved that the existence of a conic foliation on X is equivalent to the existence of a conic fibration on a birational model of X. This is done by the technique of S. Mori on lifting of rational curves in characteristic p to characteristic 0. [S. Mori, Ann. Math., II. Ser. 110, 593-606 (1979; Zbl 0423.14006)].

The author studies algebraic smooth foliations on smooth schemes X. They are by definition the involutive subbundles of the tangent sheaf \(T_ X\), which are integrable. The integrability is defined in a usual manner when the base scheme contains \({\mathbb{Q}}\) while, if the base field is of positive characteristic, then it is defined along with a lift to characteristic 0. The author studies a holomorphic map \(f: Y\to X,\) with a smooth foliation \({\mathcal F}\) on X (proper smooth, over characteristic 0), and considers deformations of f along the foliation, i.e. those deformations whose infinitesimal deformations are in the pull back of \({\mathcal F}\). The existence of universal formal deformations is proved along the line of Grothendieck’s SGA (Sémin. géométrie algébrique). As an application, it is proved that the existence of a conic foliation on X is equivalent to the existence of a conic fibration on a birational model of X. This is done by the technique of S. Mori on lifting of rational curves in characteristic p to characteristic 0. [S. Mori, Ann. Math., II. Ser. 110, 593-606 (1979; Zbl 0423.14006)].

Reviewer: E.Horikawa

##### MSC:

14D15 | Formal methods and deformations in algebraic geometry |