Exceptional subvariety

A closed subvariety $ Y $ of an algebraic variety $ X $, defined over an algebraically closed field, that can be mapped onto a subvariety $ Y _ {1} $ of lesser dimension by a proper birational morphism $ f : X \rightarrow X _ {1} $ such that $ f : X \setminus Y \rightarrow X _ {1} \setminus f ( X) $ is an isomorphism (cf. also Proper morphism). The morphism $ f $ is called a contraction of $ Y $ onto $ Y _ {1} = f ( Y) $; this concept is a particular case of that of a modification of algebraic spaces [3]. If $ X $, $ X _ {1} $, $ Y $, $ Y _ {1} $ are smooth irreducible varieties, $ Y $ is called an exceptional subvariety of the first kind. If $ Y $ has codimension 1 in $ X $, 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 $ Y $ on a smooth surface $ X $ is an exceptional subvariety of the first kind if and only if it is isomorphic to the projective line $ P ^ {1} $ and if its self-intersection index $ ( Y \cdot Y ) $ on $ X $ is equal to $ - 1 $( cf. [1], [9]). This criterion can be generalized to one-dimensional subschemes of a two-dimensional regular scheme (cf. [6], [10]). If $ Y = \sum _ {i=} 1 ^ {m} Y _ {i} $ is an arbitrary connected complete curve with irreducible components $ Y _ {i} $ on a smooth projective surface $ X $, then a necessary (but not sufficient) condition for $ Y $ to be exceptional is that the matrix $ ( Y _ {i} \cdot Y _ {j} ) $ 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 $ Y $ in a smooth algebraic variety $ X $ is exceptional of the first kind relative to a contraction to a point if the following two conditions hold: a) $ Y \cong P ^ {r} $, where $ r = \mathop{\rm dim} X - 1 $; and b) the normal bundle $ N _ {Y / X } $ to $ Y $ in $ X $ is defined by a divisor $ - H $, where $ H $ is a hyperplane in $ P ^ {r} $. In this case $ X _ {1} $ is projective. The corresponding contraction $ f $ is a monoidal transformation with centre at the point $ f ( Y) $( cf. [7], [8]).

In the analytic case, necessary and sufficient conditions for a connected compact complex submanifold $ Y $ in a complex manifold $ X $ to be exceptional of the first kind have been found; the corresponding contraction $ f $ must be a monoidal transformation with centre at $ Y _ {1} = f ( Y) $( cf. Exceptional analytic set). For algebraic varieties the corresponding conditions are necessary, but not always sufficient.

For a contraction $ f : X \rightarrow X _ {1} $ of an exceptional subvariety of the first kind $ Y $ in a projective algebraic variety $ X $ onto a zero-dimensional subvariety $ Y _ {1} $ in an algebraic variety $ X _ {1} $, $ X _ {1} $ need not be projective. Moreover, if the algebraic varieties $ X $ and $ Y $ are defined over the field of complex numbers, under an analytic contraction $ f $ of the exceptional subvariety of the first kind $ Y $, the variety $ X _ {1} $ 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 $ Y $ in a smooth algebraic space $ X $ is that the normal bundle $ N _ {Y / X } $ is negative (this condition is not sufficient for $ \mathop{\rm dim} X > 2 $). 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 $ Y $ in $ X $ is an exceptional subvariety if and only if the formal completion $ \widehat{Y} $ in $ \widehat{X} $ 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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