269 research outputs found
Nodal degenerations of plane curves and Galois covers
Globally irreducible nodes (i.e. nodes whose branches belong to the same
irreducible component) have mild effects on the most common topological
invariants of an algebraic curve. In other words, adding a globally irreducible
node (simple nodal degeneration) to a curve should not change them a lot. In
this paper we study the effect of nodal degeneration of curves on fundamental
groups and show examples where simple nodal degenerations produce
non-isomorphic fundamental groups and this can be detected in an algebraic way
by means of Galois coverings.Comment: 16 pages, 3 figure
On the Content of Polynomials Over Semirings and Its Applications
In this paper, we prove that Dedekind-Mertens lemma holds only for those
semimodules whose subsemimodules are subtractive. We introduce Gaussian
semirings and prove that bounded distributive lattices are Gaussian semirings.
Then we introduce weak Gaussian semirings and prove that a semiring is weak
Gaussian if and only if each prime ideal of this semiring is subtractive. We
also define content semialgebras as a generalization of polynomial semirings
and content algebras and show that in content extensions for semirings, minimal
primes extend to minimal primes and discuss zero-divisors of a content
semialgebra over a semiring who has Property (A) or whose set of zero-divisors
is a finite union of prime ideals. We also discuss formal power series
semirings and show that under suitable conditions, they are good examples of
weak content semialgebras.Comment: Final version published at J. Algebra Appl., one reference added,
three minor editorial change
About multiplicities and applications to Bezout numbers
Let denote a local Noetherian ring and
an ideal such that for a
finitely generated -module . Let \au = a_1,\ldots,a_d denote a system
of parameters of such that for . It follows that \chi := e_0(\au;M)
- c \cdot e_0(\mathfrak{q};M) \geq 0, where .
The main results of the report are a discussion when resp. to
describe the value of in some particular cases. Applications concern
results on the multiplicity e_0(\au;M) and applications to Bezout numbers.Comment: 11 pages, to appear Springer INdAM-Series, Vol. 20 (2017
Some closure operations in Zariski-Riemann spaces of valuation domains: a survey
In this survey we present several results concerning various topologies that
were introduced in recent years on spaces of valuation domains
Rank one discrete valuations of power series fields
In this paper we study the rank one discrete valuations of the field
whose center in k\lcor\X\rcor is the maximal ideal. In
sections 2 to 6 we give a construction of a system of parametric equations
describing such valuations. This amounts to finding a parameter and a field of
coefficients. We devote section 2 to finding an element of value 1, that is, a
parameter. The field of coefficients is the residue field of the valuation, and
it is given in section 5.
The constructions given in these sections are not effective in the general
case, because we need either to use the Zorn's lemma or to know explicitly a
section of the natural homomorphism R_v\to\d between the ring and
the residue field of the valuation .
However, as a consequence of this construction, in section 7, we prove that
k((\X)) can be embedded into a field L((\Y)), where is an algebraic
extension of and the {\em ``extended valuation'' is as close as possible to
the usual order function}
Generic coverings of plane with A-D-E-singularities
We generalize results of the paper math.AG/9803144, in which Chisini's
conjecture on the unique reconstruction of f by the curve B is investigated.
For this fibre products of generic coverings are studied. The main inequality
bounding the degree of a covering in the case of existence of two nonequivalent
coverings with the branch curve B is obtained. This inequality is used for the
proof of the Chisini conjecture for m-canonical coverings of surfaces of
general type for .Comment: 43 pages, 20 figures; to appear in Izvestiya Mat
Formal groups arising from formal punctured ribbons
We investigate Picard functor of a formal punctured ribbon. We prove that
under some conditions this functor is representable by a formal group scheme.
Formal punctured ribbons were introduced in arXiv:0708.0985.Comment: 42 pages, minor change
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