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The goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry. [ABSTRACT FROM AUTHOR]

This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the Jackson algebra, that will be used in sequels to this paper to study non-commutative arithmetic geometry. In this paper this algebra is studied from a ring-theoretic and geometric viewpoint. Among other things it turns out that this algebra is a "non-commutative family" of central simple algebras and thus parameterizes Brauer classes over extensions of the base. [ABSTRACT FROM AUTHOR]

We study the following problem in computer vision from the perspective of algebraic geometry: Using m pinhole cameras we take m pictures of a line in P 3 . This produces m lines in P 2 and the question is which m-tuples of lines can arise that way. We are interested in polynomial equations and therefore study the complex Zariski closure of all such tuples of lines. The resulting algebraic variety is a subvariety of (P 2) m and is called line multiview variety. In this article, we study its ideal. We show that for generic cameras the ideal is generated by 3 × 3 -minors of a specific matrix. We also compute Gröbner bases and discuss to what extent our results carry over to the non-generic case. [ABSTRACT FROM AUTHOR]

Type III unprojection plays a very important role in the birational geometry of Fano threefolds (cf. Corti et al., 2000; Reid, 2000). According to Reid (2000, p. 43), it was first introduced by A. Corti on his calculations of Fano threefolds of genus 6 and 7. It seems that at present a general definition of Type III unprojection is still missing. After proving some general facts about residual ideals in Section 2, we propose a definition for the generic Type III unprojection (Definition 3.3), and prove in Theorem 3.5 that it gives a Gorenstein ring. Communicated by W. Bruns [ABSTRACT FROM AUTHOR]

The purpose of this article is to give criteria for the Gorenstein property of the blowup of the spectrum of a Gorenstein local ring A along an ideal I⊂⃒A by means of the coefficient ideal of I. Moreover, we investigate the conductor ideal and the Gorenstein property of the fiber cone. [ABSTRACT FROM AUTHOR]

We list all possible configurations of singular fibers on rational elliptic surfaces in characteristic three and prove that no others exist. In total, we find 267 distinct configurations. Configurations of purely multiplicative fibers in characteristic three are compared to those in characteristics zero and two. [ABSTRACT FROM AUTHOR]

The Hilbert scheme of 3-folds in ℙ n , n ≥ 6 , that are scrolls over ℙ 2 or over a smooth quadric surface Q ⊂⃒ ℙ 3 or that are quadric or cubic fibrations over ℙ 1 is studied. All known such threefolds of degree 7 ≤ d ≤ 11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed. [ABSTRACT FROM AUTHOR]

This article provides new, relatively simple proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive the description of the natural partial order of unipotent classes in the general linear group in terms of their Jordan blocks. This result was originally due to Gerstenhaber. Finally, we obtain a proof of the Bala-Carter Theorem, which holds even in certain bad characteristics. This proof requires the prior knowledge of the number of unipotent classes, unlike the original proofs due to Bala, Carter and Pommerening. [ABSTRACT FROM AUTHOR]

In this article, using the local parametric equations of a generic projection π of a smooth projective variety X , at an analytically irreducible singular point y of X ′ = π( X ), the defining ideals J and J ′ of X ′ and its singular locus at y are expressed as ideals of maximal and sub-maximal minors of certain Sylvester matrix 𝒬. The proof is obtained by a convenient reduction of 𝒬 to a “generic pluri-circulant matrix” P and the construction of minimal Gröbner bases for the ideal of t -minors of P and for the ideals J and J ′. The depth of local rings of X ′ and Sing( X ′) at y are also computed in terms of the multiplicity at y . [ABSTRACT FROM AUTHOR]

For projective codimension two surfaces and threefolds whose singular locus is one dimensional, we get the sharp Castelnuovo-Mumford regularity bound in terms of degrees of defining equations and give the classification of nearly extremal cases. This is a generalization of the result of Bertram et al. [ABSTRACT FROM AUTHOR]

Using the local subgroup strategy of An and O'Brien (1997), An and O'Brien (1999), we classify the radical subgroups and chains of the Fischer simple group Fi 22 and verify the Alperin weight conjecture and the Uno reductive conjecture for this group; the latter is a refinement of the Dade reductive and Isaacs–Navarro conjectures. [ABSTRACT FROM AUTHOR]

Let X = Spec ( R ) be a reduced equidimensional algebraic variety over an algebraically closed field k . Let Y = Spec ( R /??) be a codimension one ordinary multiple subvariety, where ?? is a prime ideal of height 1 of R . If U is a nonempty open subset of Y and ?? a closed point of U , we denote by A ? R ?? its local ring in X , by ?? the extension of ?? in A , and by K the algebraic closure of the residue field k (??). Then there exists a bijection ? ?? : Proj ( G ?? ( A ) ? A /?? k ) ? Proj ( G ( A ?? ) ? k (?? ) K ) such that for every subset S of Proj ( G ?? ( A ) ? A /?? k ), the Hilbert function of S coincides with the Hilbert function of ? ?? (S). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety. [ABSTRACT FROM AUTHOR]

The sheaf of differential operators on log schemes is defined and studied. Then logarithmic differential operators on compactified Jacobians of singular curves are studied. In particular, a logarithmic heat equation for theta functions is produced geometrically. [ABSTRACT FROM AUTHOR]

In Keilen (2003) we gave sufficient conditions for the irreducibility of the family Virr([This symbol cannot be presented in ASCII format]l,...,[This symbol cannot be presented in ASCII format]r) of irreducible curves in the linear system \D\t with precisely r singular points of topological, respectively, analytical, types [This symbol cannot be presented in ASCII format]l,... ,[This symbol cannot be presented in ASCII format]r on several classes of smooth projective surfaces Σ. The conditions were of the form [This symbol cannot be presented in ASCII format] where τ* is some invariant of singularity types, KΣ is the canonical divisor of Σ, and γ is some constant. In the present paper we improve this condition, that is, the constant γ, by a factor of 9. [ABSTRACT FROM AUTHOR]

L-functors (Rump, 2001) provide a new tool for the study of Auslander--Reiten quivers associated with an isolated singularity in the sense of M. Auslander. We show that L-functors L, L- : M → M admit an intrinsic definition for an arbitrary additive category M. When they exist, they endow M with a structure closely related to that of a triangulated category. If M is the homotopy category M(&scriptA; ) of two-termed complexes over an additive category &scriptA;, we establish a one-to-one correspondence between L-functors on M(&scriptA;) and classes of short exact sequences in &scriptA; which make &scriptA; into an exact category with almost split sequences. This applies, in particular, to categories, &scriptA; = Λ-CM of Cohen--Macaulay modules over a Cohen--Macaulay R-order Λ for arbitrary dimension of R. [ABSTRACT FROM AUTHOR]

On vérifie, en degré d ≤ 11, la conjecture suivante:“le schéma de Hilbert des courbes lisses connexes de degré d et de genre g de l'espace projectifá trois dimensions sur le corps des complexes est irréductible pour g ≤ 2d − 9” ABSTRACT We check that the following conjecture holds true for d ≤ 11 :“the Hilbert scheme of smooth and connected space curves of degree d and genus g is irreducible provided that g ≤ 2d − 9.” [ABSTRACT FROM AUTHOR]

Let G ⊂ SO(4) denote a finite subgroup containing the Heisenberg group. In this paper we classify all such groups, we find the dimension of the spaces of G-invariant polynomials and we give equations for the generators whenever the space has dimension two. Then we complete the study of the corresponding G-invariant pencils of surfaces in [Popf]3 which we started in Sarti [Sarti, A. (2000). Pencils of symmetric surfaces in [Popf]3(C). J. Algebra 246:429–452]. It turns out that we have five more pencils, two of them containing surfaces with nodes. [ABSTRACT FROM AUTHOR]

Let G be a finite group and let cd(G) be the set of irreducible character degrees of G. The degree graph Δ(G) of G is the graph whose set of vertices is the set of primes that divide degrees in cd(G), with an edge between p and q if pq divides a for some degree a ϵ cd(G). In this paper, we determine the graph Δ(G) when G is a finite simple group of exceptional Lie type. [ABSTRACT FROM AUTHOR]

We study the automorphism and collineation groups of plane curves, i.e., in IP², that are not necessarily smooth, and obtain bounds for these curves in terms of, their degree and the number of singularities. We also introduce the notion of bad curves and good curves, and show that all bad curves are rational. [ABSTRACT FROM AUTHOR]

In this note, we classify all the polarized Fano threefold (X, H) with BsH ≠ &0slash;. As corollaries we obtained that (1) the very ample part of the conjecture of Fujita holds for smooth Fano threefolds and (2) global Seshadri constants of ample divisors on Fano threefolds are bounded from below by 1 except three types of polarized Fano threefolds. [ABSTRACT FROM AUTHOR]

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps An → Am in a class of commutative Von Neumann regular rings including p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e., sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to the study of a class of Boolean metric spaces, that we call here CFG-spaces. [ABSTRACT FROM AUTHOR]

In this article we prove Broué's conjecture on isotypies for the case of principal blocs of Sp(4, Fq)〈σ〉 : a split cyclic extension of the symplectic group of dimension 4 over a finite field Fq for q odd. The core of the proof is to lift to the group Sp(4, Fq)〈σ〉 an isotypy of the group Sp(4, Fq). For this purpose, we build a convenient version of a theorem of lifting isotypies. [ABSTRACT FROM AUTHOR]

We prove that there exists an integer M such that if S ⊂ [Popf]4 is a smooth surface with deg(S) > M, then the canonical map of S is birational. Then we consider surfaces, S, satisfying hi([Iscr]S(3 − i)) = 0, 0 ≤ i ≤ 2 and show that they are regular and that their canonical system is base point free and very ample if deg(S) > M and S doesn't contain −2-curves. [ABSTRACT FROM AUTHOR]

The concept of a composed product for univariate polynomials has been explored oextensively by Brawley, Brown, Carlitz, Gao, Mills et al. Starting with these fundamental ideas and using fractional power series representation of bivariate polynomials, the current authors Mills and Neuerburg [Mills, D., Neuerburg, K. M. A bivariate analogue to the composed product of polynomials. Algebra Colloquium (to appear)] generalize the univariate results by defining and investigating a bivariate composed sum, composed multiplication, and composed product (based on function composition) for certain classes of bivariate polynomials. In this, the sequel, we extend the generalizations to certain classes of multivariate polynomials, written as f(x1,…,xn), over an algebraically closed field of characteristic zero. In particular, we consider a composed sum and composed multiplication defined on the class of quasiordinary polynomials. We then consider what algebraic structure is imposed by these operations. Next, we consider a generalization of the geometry associated with the class of ν-quasiordinary polynomials. Finally, we utilize this geometry to show that an algorithm given by Gao and Lauder [Gao, S., Lauder, A. G. B. (2001). Decomposition of polytopes and polynomials. Discrete Comput. Geom. 26(1):89–104] to determine whether a given bivariate polynomial is absolutely irreducible over a given field can be used to ascertain whether a given trivariate polynomial that factors completely in one variable decomposes according to the composed multiplication operation. [ABSTRACT FROM AUTHOR]

Recently Jain, López-Permouth and Syed classified indecomposable non-local right continuous right π-rings. In the present article we describe the structure of right continuous right π-rings. [ABSTRACT FROM AUTHOR]

In this note we prove that a smooth order satisfying the reverse geometric engineering conditions in stringtheory of Berenstein (Berenstein, D. Reverse geometric engineering of singularities, arXiv:hep-th/0201093, Sec. 2) (in any compactifying dimension) is Auslander regular. Moreover, we classify the étale local structure of smooth orders over an isolated central singularity. [ABSTRACT FROM AUTHOR]

Based on tilting theory, we demonstrate the existence of homogeneous deformations for the Brauer tree algebras, which are derived naturally from a tilting complex. [ABSTRACT FROM AUTHOR]

We outline the results of our revisiting Hermann Schubert's work on the enumerative geometry of cuspidal cubics in P[SUP3](Sec. 23 of his Kalkül der abzählenden Geometrie, Teubner, 1879. Rep. in 1979 by Springer-Verlag). There are three main aspects that we would like to point to. First, we describe the spaces parameterizing cuspidal cubics in P[SUP3], as well as several di3erent degenerations, using modern algebraic geometry language and techniques. Then we get formulas, by means of today's intersection theory, for the relevant relations among conditions and degenerations, and for all the intersection numbers in which Schubert was in principle interested. And finally there is the computational aspect, which has been an adventure on its own: the computations have been performed by means of the mathematical computation system OMEGAMATH, together with the WIT module. They are discussed briefly in the final Section, with references to detailed information, and here we would just like to say that one of our motivations has been to test that system in what has turned out to be an interesting computational project. Our final table for the cuspidal cubics, which has the 19778 nonzero numbers involving the nine first-order conditions considered by Schubert, fully confirms the fraction of numbers computed by Schubert, as listed on pages 140-143 of the Kalkül. (For those interested in getting our table, please see the indications in the last section of the full paper.) For an assessment of whether or not the numbers computed by Schubert are fully representative of the problems involved in computing all of them, see the Remark at the end of Sec. 3. [ABSTRACT FROM AUTHOR]

We introduce effective loci, which are central for the study of differential systems, in a very general setting. The greater generality makes them useful in a wider range of situations, and it also makes their study more straightforward and convenient technically. Our treatment of effective loci is based on families of relative tangents. Such families give rise, in natural ways, to Semple sheaves, and hence allow us to construct the higher contact loci by straightforward iteration. Our main result is a local description of generic effective loci, on key Grassmannians for geometric applications. We show, by means of very natural frames, that these effective loci are precisely where certain generic matrices become symmetric. [ABSTRACT FROM AUTHOR]

This paper deals with the classification of plane algebroid branches, modulo contact equivalence, within the equisingularity class determined by the semigroup of values generated by the integers 6, 9 and 19. This illustrates how one can use the techniques developed in Hefez and Hernandes (Hefez, A., Hernandes, M. E. (2001). Computational Methods in the Local Theory of Curves. 23° Colóquio Brasileiro de Matemática. Rio de Janeiro: Publicações Matemáticas, IMPA), to make computations in the module of Kähler differentials of the local ring of an algebroid branch. [ABSTRACT FROM AUTHOR]

We study the set of rational curves of a certain topological type in general members of certain families of Calabi-Yau threefolds. For some families we investigate to what extent it is possible to conclude that this set is finite. For other families we investigate whether this set contains at least one point representing an isolated rational curve. Our study is inspired by Johnsen and Kleiman (Johnsen, T., Kleiman, S. (1996). Rational Curves of degree at most 9 on a general quintic threefold. [ABSTRACT FROM AUTHOR]

Given 5 points in 3-space and two snapshots of these points, taken with a camera at two different positions. Then, in general, there are 10 possibilities for the second position of the camera relative to its first position. The result is well known. We prove it using the Thom-Porteous Formula. [ABSTRACT FROM AUTHOR]

In this article we classify all the smooth threefolds in [Popf]5 with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties in [Popf]n with one apparent (n - 1)-point and with irreducible family of (n - 1)-secant lines. [ABSTRACT FROM AUTHOR]

Lie quasi-bialgebras (also called Jacobian quasi-bialgebras) are generalisations of Lie bialgebras while di3erential quasi-Batalin-Vilkovisky algebras are generalisations of di3erential Batalin-Vilkovisky algebras. We show that the kernel of the Laplacian operator of a quasi-Batalin-Vilkovisky algebra is a homotopy BV-algebra (BV[SUB∞]-algebra). We establish a one-to-one correspondence between the Lie quasi-bialgebra structures on a finite-dimensional vector space with a generalized co-character and differential quasi-Batalin-Vilkovisky structures on the exterior algebra of the dual vector space. [ABSTRACT FROM AUTHOR]

All solutions of Pythagorean equation (P-equation) x²[sub 1] + x²[sub 2] = x²[sub 3] in relatively free rings of varieties of n-nilpotent associative or commutative-associative rings (n = 3,4) are described. In particular, it is shown that Pythagorean equation has no minimal and complete set of solutions in free rings of such varieties. This implies that unification type of these varieties is nullary. It is shown that unification type of P-equation in varieties of associative rings and commutativeassociative rings without unit is not finitary. Hence, the unification type of these varieties is also not finitary. We show also that the variety of commutative-associative 3 (or 4)nilpotent rings of characteristic 2 has nullary unification type. [ABSTRACT FROM AUTHOR]

We introduce the analytification of a scheme locally of finite type over . On one has an exponential sequence if is reduced. This exponential sequence gives rise to an analytic description of the Picard group Pic() if is proper and reduced. Using this description we generalize Harnack's Inequality for real algebraic curves to arbitrary proper reduced real schemes. [ABSTRACT FROM AUTHOR]

Some observations and conjectures concerning finite dimensional associative commutative algebras that are degenerations of semisimple algebras. “…on sait qu'on ne sait rien sur cette sorte d'algèbre.” André Weil [ABSTRACT FROM AUTHOR]

A linear system of plane curves satisfying multiplicity conditions at points in general position is called special if the dimension is larger than the expected dimension. A (-1) curve is an irreducible curve with self intersection -1 and genus zero. The Segre-Harbourne-Hirschowitz Conjecture states that if a linear system is special then a multiple of some fixed (-1) curve is contained in every curve of the linear system. This conjecture is proven for linear systems with multiplicity four at all but one of the points. [ABSTRACT FROM AUTHOR]