Exceptional subvariety

A closed subvariety of an algebraic variety , defined over an algebraically closed field, that can be mapped onto a subvariety of lesser dimension by a proper birational morphism such that is an isomorphism (cf. also Proper morphism). The morphism is called a contraction of onto ; this concept is a particular case of that of a modification of algebraic spaces [3]. If , , , are smooth irreducible varieties, is called an exceptional subvariety of the first kind. If has codimension 1 in , it is an exceptional divisor. Exceptional divisors on an algebraic surface are called exceptional curves.

The notion of an exceptional subvariety can be naturally generalized to schemes, algebraic and complex-analytic spaces. The corresponding morphisms are called contractions; the notion of an exceptional subvariety of the first kind can also be naturally generalized. An exceptional subvariety in a complex-analytic space is also called an exceptional analytic set.

Characterizing the exceptional subvarieties within the ambient variety is one of the basic problems in birational geometry. Historically, the first example of such a characterization is the Enriques–Castelnuovo criterion: An irreducible complete curve on a smooth surface is an exceptional subvariety of the first kind if and only if it is isomorphic to the projective line and if its self-intersection index on is equal to (cf. [1], [9]). This criterion can be generalized to one-dimensional subschemes of a two-dimensional regular scheme (cf. [6], [10]). If is an arbitrary connected complete curve with irreducible components on a smooth projective surface , then a necessary (but not sufficient) condition for to be exceptional is that the matrix is negative definite (cf. [2]). In the case of a connected complex curve on a smooth complex surface, or a connected complete curve on a smooth two-dimensional algebraic space, the analogous condition is necessary and sufficient for exceptionality.

The multi-dimensional analogue of the Enriques–Castelnuovo criterion for contractions to a point has the following form [5]: An irreducible complete subvariety in a smooth algebraic variety is exceptional of the first kind relative to a contraction to a point if the following two conditions hold: a) , where ; and b) the normal bundle to in is defined by a divisor , where is a hyperplane in . In this case is projective. The corresponding contraction is a monoidal transformation with centre at the point (cf. [7], [8]).

In the analytic case, necessary and sufficient conditions for a connected compact complex submanifold in a complex manifold to be exceptional of the first kind have been found; the corresponding contraction must be a monoidal transformation with centre at (cf. Exceptional analytic set). For algebraic varieties the corresponding conditions are necessary, but not always sufficient.

For a contraction of an exceptional subvariety of the first kind in a projective algebraic variety onto a zero-dimensional subvariety in an algebraic variety , need not be projective. Moreover, if the algebraic varieties and are defined over the field of complex numbers, under an analytic contraction of the exceptional subvariety of the first kind , the variety need not be algebraic at any point, in general.

Regarding the question of contractibility of an exceptional subvariety (not necessarily of the first kind) to a point, a necessary condition of exceptionality of a complete connected algebraic subspace in a smooth algebraic space is that the normal bundle is negative (this condition is not sufficient for ). The analogous fact holds for complex spaces.

In the case of algebraic spaces, the most general criterion of exceptionality states that in the category of Noetherian algebraic spaces a subspace in is an exceptional subvariety if and only if the formal completion in is an exceptional subvariety in the category of formal algebraic spaces [3]. In other words, contraction of algebraic subspaces is possible if and only if it is possible for the corresponding formal completions.

References

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