Effective construction of an algebraic variety nonsingular in codimension one over a ground field of zero characteristic

Let k be a field of zero characteristic finitely generated over a primitive subfield. Let f be a polynomial of degree at most d in n variables, with coefficients from k, irreducible over an algebraic closure [`(k)] \bar{k} . Then we construct an algebraic variety V nonsingular in codimension one and a finite birational isomorphism V → Z(f), where Z(f) is the hypersurface of all common zeros of the polynomial f in the affine space. The running time of the algorithm for constructing V is polynomial in the size of the input. Bibliography: 8 titles.     

The index of an algebraic variety

Let K be the field of fractions of a Henselian discrete valuation ring  ${{\mathcal {O}}_{K}}$ . Let X _K /K be a smooth proper geometrically connected scheme admitting a regular model $X/{{\mathcal {O}}_{K}}$ . We show that the index δ(X _K /K) of X _K /K can be explicitly computed using data pertaining only to the special fiber X _k /k of the model X. We give two proofs of this theorem, using two moving lemmas. One moving lemma pertains to horizontal 1-cycles on a regular projective scheme X over the spectrum of a semi-local Dedekind domain, and the second moving lemma can be applied to 0-cycles on an $\operatorname {FA} $ -scheme X which need not be regular. The study of the local algebra needed to prove these moving lemmas led us to introduce an invariant γ(A) of a singular local ring $(A, {\mathfrak {m}})$ : the greatest common divisor of all the Hilbert-Samuel multiplicities e(Q,A), over all ${\mathfrak {m}}$ -primary ideals Q in ${\mathfrak {m}}$ . We relate this invariant γ(A) to the index of the exceptional divisor in a resolution of the singularity of $\operatorname {Spec}A$ , and we give a new way of computing the index of a smooth subvariety X/K of ${\mathbb{P}}^{n}_{K}$ over any field K, using the invariant γ of the local ring at the vertex of a cone over X.     

A variety of semigroups without an irreducible basis for identities

An example of a variety of semigroups not having an irreducible basis for identities is given. Also a variety of semigroups, the lattice of subvarieties of which contains a nonidentity element without any coverings, is constructed.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 865–872, June, 1977.     

Any algebraic variety in positive characteristic admits a projective model with an inseparable Gauss map

We determine the values attained by the rank of the Gauss map of a projective model for a fixed algebraic variety in positive characteristic p. In particular, it is shown that any variety in p>0 has a projective model such that the differential of the Gauss map is identically zero. On the other hand, we prove that there exists a product of two or more projective spaces admitting an embedding into a projective space such that the differential of the Gauss map is identically zero if and only if p=2.     

On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan

We deal with the construction of sequences of irreducible polynomials with coefficients in finite fields of even characteristic. We rely upon a transformation used by Kyuregyan in 2002, which generalizes the Q-transform employed previously by Varshamov and Garakov (1969) as well as by Meyn (1990) for the synthesis of irreducible polynomials. While in the iterative procedure described by Kyuregyan the coefficients of the initial polynomial of the sequence have to satisfy certain hypotheses, in the present paper these conditions are removed. We construct infinite sequences of irreducible polynomials of nondecreasing degree starting from any irreducible polynomial.     

Bounds for the greatest characteristic root of an irreducible nonnegative matrix

A new lower bound for the Perron root for irreducible, non-negative matrices is obtained which is, in particular, a better bound than the Frobenius bound [w = max(a_kk)] if all the main diagonal elements are zero.     

Numerical computation of the genus of an irreducible curve within an algebraic set

The common zero locus of a set of multivariate polynomials (with complex coefficients) determines an algebraic set. Any algebraic set can be decomposed into a union of irreducible components. Given a one-dimensional irreducible component, i.e. a curve, it is useful to understand its invariants. The most important invariants of a curve are the degree, the arithmetic genus and the geometric genus (where the geometric genus denotes the genus of a desingularization of the projective closure of the curve). This article presents a numerical algorithm to compute the geometric genus of any one-dimensional irreducible component of an algebraic set.     

On the topology of the joined point-pairs of an algebraic variety

Sunto Data una varietà complessa non singolare V, si può definire unacoppia congiunta di punti di V. Nel caso ove V è una varietà razionale, di un tipo molto ristretto (§ 2), si considerano le proprietà topologiche dell'insieme di coppie congiunte. Questo è uno spazio fibrato, il che rende possibile la determinazione dei suoi gruppi di omologia. Si costruisce esplicitamente una base per questi gruppi.