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 $(A,\mathfrak{m},\Bbbk)$ denote a local Noetherian ring and $\mathfrak{q}$ an ideal such that $\ell_A(M/\mathfrak{q}M) < \infty$ for a finitely generated $A$-module $M$. Let \au = a_1,\ldots,a_d denote a system of parameters of $M$ such that $a_i \in \mathfrak{q}^{c_i} \setminus \mathfrak{q}^{c_i+1}$ for $i=1,\ldots,d$. It follows that \chi := e_0(\au;M) - c \cdot e_0(\mathfrak{q};M) \geq 0, where $c = c_1\cdot \ldots \cdot c_d$. The main results of the report are a discussion when $\chi = 0$ resp. to describe the value of $\chi$ 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 $k((X_1,..., X_n))$ 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 $\sigma$ of the natural homomorphism R_v\to\d between the ring and the residue field of the valuation $v$. However, as a consequence of this construction, in section 7, we prove that k((\X)) can be embedded into a field L((\Y)), where $L$ is an algebraic extension of $k$ 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 $m\ge 5$.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