Birationality of étale maps via surgery
نویسندگان
چکیده
We use a counting argument and surgery theory to show that if D is a sufficiently general algebraic hypersurface in C, then any local diffeomorphism F : X → C of simply connected manifolds which is a d-sheeted cover away from D has degree d = 1 or d = ∞ (however all degrees d > 1 are possible if F fails to be a local diffeomorphism at even a single point). In particular, any étale morphism F : X → C of algebraic varieties which covers away from such a hypersurface D must be birational.
منابع مشابه
Linear syzygies and birational combinatorics
Let F be a finite set of monomials of the same degree d ≥ 2 in a polynomial ring R = k[x1, . . . , xn] over an arbitrary field k. We give some necessary and/or sufficient conditions for the birationality of the ring extension k[F ] ⊂ R, where R is the dth Veronese subring of R. One of our results extends to arbitrary characteristic, in the case of rational monomial maps, a previous syzygy-theor...
متن کاملÉtale Covers of Affine Spaces in Positive Characteristic
We prove that every projective variety of dimension n over a field of positive characteristic admits a morphism to projective n-space, étale away from the hyperplane H at infinity, which maps a chosen divisor into H and a chosen smooth point not on the divisor to some point not in H .
متن کاملThe dynamical Mordell-Lang problem for étale maps
We prove a dynamical version of the Mordell-Lang conjecture for étale endomorphisms of quasiprojective varieties. We use p-adic methods inspired by the work of Skolem, Mahler, and Lech, combined with methods from algebraic geometry. As special cases of our result we obtain a new proof of the classical Mordell-Lang conjecture for cyclic subgroups of a semiabelian variety, and we also answer posi...
متن کاملMore Étale Covers of Affine Spaces in Positive Characteristic
We prove that every geometrically reduced projective variety of pure dimension n over a field of positive characteristic admits a morphism to projective n-space, étale away from the hyperplane H at infinity, which maps a chosen divisor into H and some chosen smooth points not on the divisor to points not in H . This improves an earlier result of the author, which was restricted to infinite perf...
متن کاملOn the homology of two-dimensional elimination
We study birational maps with empty base locus defined by almost complete intersection ideals. Birationality is shown to be expressed by the equality of two Chern numbers. We provide a relatively effective method of their calculation in terms of certain Hilbert coefficients. In dimension two, after observing that the structure of its irreducible ideals (always complete intersections by a classi...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006