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Suppose that (K, $ u$) is a valued field, f (z) $in$ K[z] is a unitary and irreducible polynomial and (L, $omega$) is an extension of valued fields, where L = K[z]/(f (z)). Further suppose that A is a local domain with quotient field K such that $ u$ has nonnegative value on A and positive value on its maximal ideal, and that f (z) is in A[z]. This paper is devoted to the problem of describing the structure of the associated graded ring gr $omega$ A[z]/(f (z)) of A[z]/(f (z)) for the filtration defined by $omega$ as an extension of the associated graded ring of A for the filtration defined by $ u$. In particular we give an algorithm which in many cases produces a finite set of elements of A[z]/(f (z)) whose images in gr $omega$ A[z]/(f (z)) generate it as a gr $ u$ A-algebra as well as the relations between them. We also work out the interactions of our method of computation with phenomena which complicate the study of ramification and local uniformization in positive characteristic , such as the non tameness and the defect of an extension. For valuations of rank one in a separable extension of valued fields (K, $ u$) $subset$ (L, $omega$) as above our algorithm produces a generating sequence in a local birational extension A1 of A dominated by $ u$ if and only if there is no defect. In this case, gr $omega$ A1[z]/(f (z)) is a finitely presented gr $ u$ A1-module. This is an improved version, thanks to a referees remarks.
In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being Q-Gorenstein or the pair being log Q-Gorenstein. The main features
For a reduced hypersurface $V(f) subseteq mathbb{P}^n$ of degree $d$, the Castelnuovo-Mumford regularity of the Milnor algebra $M(f)$ is well understood when $V(f)$ is smooth, as well as when $V(f)$ has isolated singularities. We study the regularity
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $xin (X,Delta)$ satisfies the ACC if the coefficients of $Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under
In this paper, we study factorizations in the additive monoids of positive algebraic valuations $mathbb{N}_0[alpha]$ of the semiring of polynomials $mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in
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