1,334 research outputs found
Asymptotic growth of saturated powers and epsilon multiplicity
Asymptotic properties of saturated powers of modules over a local domain R
are studied. Under mild conditions, it is shown that the limit as k goes to
infinity of the quotient of the saturation of the k-th power of a module E by
the k-th power of E, when divided by k^{d+e-1}, exists. Here d is the dimension
of R and e is the rank of E. We deduce that under these assumptions, the
epsilon multiplicity of E, defined by Ulrich and Validashti as a limsup,
actually exists as a limit.Comment: 9 pages. In the revised version a typo ("n" changed to "k") is fixed
in the statement of Corollary 1.3. A couple of new corollaries are added and
some references are added. In the second revision (13 pages) some extensions
from domains of depth \ge 2 are given to domains of dimension \ge
Artificial Constraints and Lipschitz Hints for Unconstrained Online Learning
We provide algorithms that guarantee regret or for online convex optimization with -Lipschitz losses for
any comparison point without prior knowledge of either or .
Previous algorithms dispense with the term at the expense of
knowledge of one or both of these parameters, while a lower bound shows that
some additional penalty term over is necessary. Previous
penalties were exponential while our bounds are polynomial in all quantities.
Further, given a known bound , our same techniques allow us to
design algorithms that adapt optimally to the unknown value of without
requiring knowledge of
Local monomialization of transcendental extensions
Suppose that f is a dominant morphism from a k-variety X to a k-variety Y,
where k is a field of characteristic 0 and v is a valuation of the function
field k(X). We allow v to be an arbitary valuation, so it may not be discrete.
We prove that there exist sequences of blowups of nonsingular subvarieties
from X' to X and from Y' to Y such that X', Y' are nonsingular and X' to Y' is
locally a monomial mapping near the center of v. This extends an earlier result
of ours (in Asterisque 260) which proves the above result with the restriction
that f is generically finite.Comment: 50 page
Counterexamples to local monomialization in positive characteristic
In this paper we consider birational properties of ramification in excellent
local rings. We give an example showing that local monomialization (and weak
local monomialization) can fail for extensions of algebraic local rings in
algebraic function fields of dimension greater than or equal to two along a
valuation over a field of positive characteristic. It was earlier proven by the
author that local monomialization holds within characteristic zero algebraic
function fields.Comment: 11 page
Strong Toroidalization of Dominant Morphisms of 3-folds
Suppose that is a dominant morphism of 3-folds over an
algebraically closed field of characteristic zero. We prove that there exist
sequences of blow ups of points and nonsingular curves and
such that the induced map is a toroidal
morphism.
This extends an earlier proof of the author of this theorem with the extra
assumption that is birational.Comment: 121 page
Rectilinearization of sub analytic sets as a consequence of local monomialization
We give a new proof of the rectilinearization theorem of Hironaka. We deduce
rectilinearization as a consequence of our theorem on local monomialization for
complex and real analytic morphisms.Comment: 13 page
Resolution of Singularities for 3-folds in positive characteristic
In this paper a concise, complete proof of resolution of singularities of
3-folds in positive characteristic (>5) is given. The first proof of this
theorem was given by Abhyankar in 1966. The resolution morphism in our proof is
an isomorphism over the nonsingular locus.Comment: 55 pages. In this final version, which is to appear in the American
Journal of Mathematics, complete details of the proof of embedded resolution
of surfaces have been added in. Two sections have been added after the
introduction, giving an outline of the proof, and discussing related results
and extensions of the material proved in this pape
Asymptotic Multiplicities
We give several new applications of our theorem on the existence of
multiplicity of graded families of ideals as a limit, including a very general
Minkowski type inequality for graded families of ideals, a very general formula
for existence of local volumes as a limit and a very general theorem on the
existence of epsilon multiplicities as a limit for modules.Comment: 29 page
Generically Finite Morphisms
We consider the problem of birationally modifying a morphism of complete
varieties to make it a morphism from a nonsingular variety to a normal variety.
Our main result is to give a counterexample to this problem. This example also
is a counterexample to the related conjecture of Abhyankar on ``weak
simultaneous global resolution''.
We also give some positive results. Forinstance, a positive result of this
kind is possible if we remove the separatedness condition
Ramification of valuations and local rings
In this paper we consider birational properties of ramification in excellent
local rings. We extend earlier results of the author with Olivier Piltant to
show that strong local monomialization is true along a valuation dominating a
defectless extension of two dimensional excellent local rings. We also obtain
general results on the structure of the extension of associated graded rings
along a valuation, and show that the invariants alpha and beta of stable forms
of two dimensional extensions in characteristic p of the author and Olivier
Piltant are not eventually constant.Comment: 31 pages. Formerly part of arXiv:1404.745
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