In this post, we introduce quantities called that are associated to smooth, complete varieties, and show that they are birational invariants. We offer two sources of motivation:
The classification of curves essentially amounts to stating that there exists one invariant (the genus), and for curves of fixed genus , there is a -dimensional family. Surfaces, however, are more complicated, and we anticipate their moduli to have more intricate data than in the case of curves; they cannot be classified by a single invariant. Without getting too far afield into a “complete” taxonomy (due to Enriques-Kodaira), we wish to find invariants of surfaces, of which the plurigenera will serve as a family of examples that straightforwardly generalize the genus of a curve. Moreover, they generalize to higher-dimensional -folds.
A question that remained open for some time was whether unirational varieties were rational – that is, if varieties admitting a dominant rational map from projective space were actually birationally equivalent to projective space. For curves, this was answered in the affirmative by a classical theorem of Luroth, and for surfaces, a result of Castelnuovo also indicated its verity. However, numerous counterexamples emerged for threefolds in the 70s. A general program, then, was to find birational invariants that took on different values for a given example when compared to .
Let be a smooth, complete variety over a field . The of are Recalling that one definition for the genus of a curve was the dimension of the space of -forms on it, we recover .
For all , the plurigenera are birational invariants for smooth complete varieties.
Hartshorne proves that the genus of a curve is a birational invariant, a proof which we shall adapt below.
Let be a birational map of smooth complete varieties. Since is normal and is complete, extends to a morphism , where is open and such that . To compare forms on to those on , we have a pullback morphism which is an isomorphism on the open subset of over which is an isomorphism. Then, taking top exterior powers and th tensor powers induces a morphism which is still generically an isomorphism. This is an injection of sheaves since both are locally free, hence torsion-free. All of this gives rise to the following commutative diagram: where the left-hand is injective since is dominant, and the right-hand is an isomorphism since , is locally free, and since is normal (and so , i.e. functions extend over codimension two or greater). Taking dimensions, it follows that for all ; replacing with the inverse of the birational map, we get that , which proves that for all , as desired.