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Register a new userOn finite determinacy of complete intersection singularities
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We give an elementary combinatorial proof of the following fact: Every real or complex analytic complete intersection germ X is equisingular -- in the sense of the Hilbert-Samuel function -- with a germ of an algebraic set defined by sufficiently long truncations of the defining equations of X.
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In this paper we give a geometric characterization of the cones of toric varieties that are complete intersections. In particular, we prove that the class of complete intersection cones is the smallest class of cones which is closed under direct sum and contains all simplex cones. Further, we show that the number of the extreme rays of such a cone, which is less than or equal to $2n-2$, is exactly $2n-2$ if and only if the cone is a bipyramidal cone, where $n>1$ is the dimension of the cone. Finally, we characterize all toric varieties whose associated cones are complete intersection cones.
We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let $N geq 2$, and consider an isolated complete intersection curve singularity germ $f colon (mathbb{C}^N,0) to (mathbb{C}^{N-1},0)$. We introduce a numerical function $m mapsto operatorname{AD}_{(2)}^m(f)$ that arises as an error term when counting $m^{mathrm{th}}$-order weight-$2$ inflection points with ramification sequence $(0, dots, 0, 2)$ in a $1$-parameter family of curves acquiring the singularity $f = 0$, and we compute $operatorname{AD}_{(2)}^m(f)$ for various $(f,m)$. Particularly, for a node defined by $f colon (x,y) mapsto xy$, we prove that $operatorname{AD}_{(2)}^m(xy) = {{m+1} choose 4},$ and we deduce as a corollary that $operatorname{AD}_{(2)}^m(f) geq (operatorname{mult}_0 Delta_f) cdot {{m+1} choose 4}$ for any $f$, where $operatorname{mult}_0 Delta_f$ is the multiplicity of the discriminant $Delta_f$ at the origin in the deformation space. Furthermore, we show that the function $m mapsto operatorname{AD}_{(2)}^m(f) -(operatorname{mult}_0 Delta_f) cdot {{m+1} choose 4}$ is an analytic invariant measuring how much the singularity counts as an inflection point. We obtain similar results for weight-$2$ inflection points with ramification sequence $(0, dots, 0, 1,1)$ and for weight-$1$ inflection points, and we apply our results to solve various related enumerative problems.
We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to numerical semigroups. This correspondence is shown to preserve the complete intersection property, and allows us to use some available algorithms to determine whether a given vanishing ideal is a complete intersection. We give formulae for the degree, and for the index of regularity of a complete intersection in terms of the Frobenius number and the generators of a numerical semigroup.
In this paper we generalize the definitions of singularities of pairs and multiplier ideal sheaves to pairs on arbitrary normal varieties, without any assumption on the variety being Q-Gorenstein or the pair being log Q-Gorenstein. The main features of the theory extend to this setting in a natural way.
We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. The generators of the toric ideal correspond to even cycles of $G$ except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of the graph satisfy the odd cycle condition. Finally we characterize all complete intersection toric ideals of graphs which are normal.
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