Edward Bierstone
Resolution of singularities of pairs preserving semi-simple normal crossings.
Fri, 09/16/2011 - 15:01 — SomNambulAuthors: Edward Bierstone, Franklin Vera Pacheco
Subjects: Algebraic Geometry
AbstractLet X denote a reduced algebraic variety and D a Weil divisor on X. The pair
(X,D) is said to be semi-simple normal crossings (semi-snc) at a point a of X
if X is simple normal crossings at a (i.e., a simple normal crossings
hypersurface, with respect to a local embedding in a smooth ambient variety),
and D is induced by the restriction to X of a hypersurface that is simple
normal crossings with respect to X.Resolution except for minimal singularities II. The case of four variables.
Fri, 07/29/2011 - 15:01 — SomNambulAuthors: Edward Bierstone, Pierre D. Milman, Pierre Lairez
Subjects: Algebraic Geometry
AbstractIn this sequel to Resolution except for minimal singularities I, we find the
smallest class of singularities in four variables with which we necessarily end
up if we resolve singularities except for normal crossings. The main new
feature is a characterization of singularities in four variables which occur as
limits of triple normal crossings singularities, and which cannot be eliminated
by a birational morphism that avoids blowing up normal crossings singularities.Resolution except for minimal singularities I.
Fri, 07/29/2011 - 15:01 — SomNambulAuthors: Edward Bierstone, Pierre D. Milman
Subjects: Algebraic Geometry
AbstractThe philosophy of the article is that the desingularization invariant
together with natural geometric information can be used to compute local normal
forms of singularities. The idea is used in two related problems: (1) We give a
proof of resolution of singularities of a variety or a divisor, except for
simple normal crossings (i.e., which avoids blowing up simple normal crossings,
and ends up with a variety or a divisor having only simple normal crossings
singularities). (2) For more general normal crossings (in a local analytic or
formal sense), such a result does not hold.Geometric Auslander criterion for flatness.
Fri, 09/04/2009 - 12:10 — SomNambulAuthors: Janusz Adamus, Edward Bierstone, Pierre D. Milman
Subjects: Algebraic Geometry
AbstractOur aim is to understand the algebraic notion of flatness in explicit
geometric terms. Let Y be a scheme of finite type over a perfect field, and let
f:X->Y denote a morphism of schemes that is locally of finite type. We show
that, if Y is regular, then nonflatness of f is equivalent to a severe
discontinuity of the fibres - the existence of an associated component (perhaps
embedded) at a point of the source whose image is nowhere dense in Y - after
passage to the n'th fibred-power of f, where n = dim Y.Geometric Auslander criterion for flatness of an analytic mapping.
Tue, 08/25/2009 - 22:41 — SomNambulAuthors: Janusz Adamus, Edward Bierstone, Pierre D. Milman
Subjects: Commutative Algebra
AbstractWe prove that, if F is a coherent sheaf of modules over the source of a
morphism f:X->Y of complex-analytic spaces, where Y is smooth, then the stalk
of F at a point x in X is flat over R, the local ring of the target at f(x) if
and only if the n-fold analytic tensor power of this stalk over R (where n =
dim R) has no vertical elements. The result implies that if F is a finite
module over a morphism f:X->Y of complex algebraic varieties, where Y is smooth
and n=dim Y, then the stalk of F at x is R-flat if and only if its n-fold
tensor power is a torsionfree R-module.
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