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We construct a class of projective rational varieties X of any dimension m ≥ 1, which are smooth except at a point O, with the projective space ℙ m as normalization, having smooth branches, and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sufficient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal. [ABSTRACT FROM PUBLISHER]

Let k be an algebraically closed field of characteristic 0, and let A = k[x, y]/(f) be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e., a graded A-linear homomorphism ∇: Derk(A) → Endk(M), that satisfies the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring [image omitted] admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module. [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]

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]

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]