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Let k be a field of characteristic zero. Let V be an integral affine k -variety. We prove that any torsion Kähler differential form ω ∈ Ω V / k 1 ( V ) vanishes at every formal germ of curve on V . If V is an integral hypersurface of A k N ( N ≥ 1 ), we also prove that the non-cylindrical singular points of V belong to the singular locus of any torsion Kähler differential forms. [ABSTRACT FROM AUTHOR]

Let X be a variety over a perfect field k and let X ∞ be its space of arcs. Given a closed subset Z of X , let X ∞ Z denote the subscheme of X ∞ consisting of all arcs centered at some point of Z . We prove that Local Uniformization implies that X ∞ Z has a finite number of irreducible components for each closed subset Z of X . In particular, Local Uniformization implies that X ∞ Sing X has a finite number of irreducible components. [ABSTRACT FROM AUTHOR]

For a generic f ∈ C ∞ ( R 4 , R 4 ) there is a discrete set Σ 2 ( D f ) of critical points of rank two, and there is an integer index I p ( D f ) associated to any p ∈ Σ 2 ( D f ) . We show how to compute ∑ I p ( D f ) , where p ∈ Σ 2 ( D f ) , in the case where f is a polynomial mapping. [ABSTRACT FROM AUTHOR]

To a graded finite-rank matrix factorisation of the difference of two homogeneous potentials one can assign two numerical invariants, the left and right quantum dimensions. The existence of such a matrix factorisation with non-zero quantum dimensions defines an equivalence relation between potentials, giving rise to non-obvious equivalences of categories. Restricted to ADE singularities, the resulting equivalence classes of potentials are those of type { A d − 1 } for d odd, { A d − 1 , D d / 2 + 1 } for d even but not in { 12 , 18 , 30 } , and { A 11 , D 7 , E 6 } , { A 17 , D 10 , E 7 } and { A 29 , D 16 , E 8 } . This is the result expected from two-dimensional rational conformal field theory, and it directly leads to new descriptions of and relations between the associated (derived) categories of matrix factorisations and Dynkin quiver representations. [ABSTRACT FROM AUTHOR]

An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consisting of a sheaf of ideals and a positive integer. This approach is particularly simple and seems suitable for applications to a good theory of simultaneous algorithmic resolution of singularities, specially for families parametrized by the spectrum of an artinian ring. [ABSTRACT FROM AUTHOR]

Abstract: In 2009, de Fernex and Hacon [10] proposed a generalization of the notion of the singularities to normal varieties that are not -Gorenstein. Based on their work, we generalize Kleiman’s transversality theorem to subvarieties with log terminal or log canonical singularities. We also show that some classical varieties, such as generic determinantal varieties, for general smooth curves, and certain Schubert varieties in are log terminal in de Fernex and Hacon’s notion, and canonical with some suitable boundary in the classical sense. [Copyright &y& Elsevier]

Abstract: This paper proves that the winning strategy for Hauser’s version of Hironaka’s polyhedra game is almost arbitrary. The winning strategy and its associated invariants are based on an algorithm of matrix triangulations and matrix diagonalizations. It is proved that if a set sequence constitutes a winning strategy for the game, then so does every set sequence containing it. The same holds for Hironaka’s version of the game if every move is permissible. [Copyright &y& Elsevier]

Abstract: Let be a regular noetherian local ring of dimension and be a sequence of successive quadratic transforms along a regular prime ideal of (i.e if is the strict transform of in , then , ). We say that is maximal for if for every non-negative integer and for every prime ideal of such that is a quadratic sequence along with , we have . We show that is maximal for if and only if is a valuation ring of dimension one. In this case, the equimultiple locus at is the set of elements of the maximal ideal of for which the multiplicity is stable along the sequence , provided that the series of real numbers given by the multiplicity sequence associated with diverges. Furthermore, if we consider an ideal of , we also show that is normally flat along at the closed point if and only if the Hironaka’s character is stable along the sequence . This generalizes well known results for the case where has height one (see [B.M. Bennett, On the characteristic functions of a local ring, Ann. of Math. Second Series 91 (1) (1970) 25–87]). [Copyright &y& Elsevier]

Abstract: We give an algebraic proof of a theorem of H. Maugendre showing how the jacobian quotients of a pair of germs of plane curve may be computed from their simultaneous immerged resolution, thus proving in particular their topological invariance. [Copyright &y& Elsevier]