scholarly journals On the topology of full non-degenerate complete intersection variety

1991 ◽  
Vol 121 ◽  
pp. 137-148 ◽  
Author(s):  
Mutsuo Oka
Keyword(s):  
Complete Intersection ◽  
Exceptional Divisor ◽  
Isolated Singularity ◽  
Laurent Polynomials ◽  
Intersection Variety

Let h1(u),…, hk(u) be Laurent polynomials of m-variables and letbe a non-degenerate complete intersection variety. Such an intersection variety appears as an exceptional divisor of a resolution of non-degenerate complete intersection varieties with an isolated singularity at the origin (Ok4]).

scholarly journals NOETHER–LEFSCHETZ THEORY AND NÉRON–SEVERI GROUP

2012 ◽  
Vol 23 (01) ◽  
pp. 1250004 ◽  
Author(s):  
VINCENZO DI GENNARO ◽  
DAVIDE FRANCO
Keyword(s):  
Complete Intersection ◽  
Large Degree ◽  
Closed Subscheme ◽  
Smooth Hypersurface ◽  
Smooth Complex ◽  
Intersection Variety ◽  
Complex Projective

Let Z be a closed subscheme of a smooth complex projective complete intersection variety Y ⊆ ℙN, with dim Y = 2r + 1 ≥ 3. We describe the Néron–Severi group NSr(X) of a general smooth hypersurface X ⊂ Y of sufficiently large degree containing Z.

scholarly journals The Bruce–Roberts Number of A Function on A Hypersurface with Isolated Singularity

2020 ◽  
Vol 71 (3) ◽  
pp. 1049-1063
Author(s):  
J J Nuño-Ballesteros ◽  
B Oréfice-Okamoto ◽  
B K Lima-Pereira ◽  
J N Tomazella
Keyword(s):  
Complete Intersection ◽  
Hypersurface Singularity ◽  
Isolated Singularity ◽  
Characteristic Variety ◽  
Intersection Singularity ◽  
Complete Intersection Singularity ◽  
Isolated Complete Intersection Singularity

Abstract Let $(X,0)$ be an isolated hypersurface singularity defined by $\phi \colon ({\mathbb{C}}^n,0)\to ({\mathbb{C}},0)$ and $f\colon ({\mathbb{C}}^n,0)\to{\mathbb{C}}$ such that the Bruce–Roberts number $\mu _{BR}(f,X)$ is finite. We first prove that $\mu _{BR}(f,X)=\mu (f)+\mu (\phi ,f)+\mu (X,0)-\tau (X,0)$, where $\mu $ and $\tau $ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen–Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.

scholarly journals HOMOLOGICAL INDEX FOR 1-FORMS AND A MILNOR NUMBER FOR ISOLATED SINGULARITIES

2004 ◽  
Vol 15 (09) ◽  
pp. 895-905 ◽  
Author(s):  
W. EBELING ◽  
S. M. GUSEIN-ZADE ◽  
J. SEADE
Keyword(s):  
Complete Intersection ◽  
Milnor Number ◽  
Isolated Singularity ◽  
Isolated Singularities ◽  
Analytic Variety ◽  
Curve Singularities ◽  
Complex Analytic Variety ◽  
Arbitrary Curve

We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by Gómez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.

scholarly journals A formula for the Euler characteristic of line singularities on singular spaces

2000 ◽  
Vol 61 (2) ◽  
pp. 335-343
Author(s):  
Guangfeng Jiang
Keyword(s):  
Complete Intersection ◽  
Euler Characteristic ◽  
Smooth Curve ◽  
Isolated Singularity ◽  
Singular Spaces ◽  
Critical Locus ◽  
Algebraic Formula ◽  
Line Singularities

We prove an algebraic formula for the Euler characteristic of the Milnor fibres of functions with critical locus a smooth curve on a space which is a weighted homogeneous complete intersection with isolated singularity.

scholarly journals On the Hodge conjecture for quasi-smooth intersections in toric varieties

2021 ◽  
Author(s):  
Ugo Bruzzo ◽  
William Montoya
Keyword(s):  
Complete Intersection ◽  
Toric Variety ◽  
Toric Varieties ◽  
Picard Group ◽  
Hodge Conjecture ◽  
Smooth Hypersurface ◽  
Suitable Notion

AbstractWe establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety $${{\mathbb {P}}}_\Sigma ^{2k+1}$$ P Σ 2 k + 1 has Picard group $${\mathbb {Z}}$$ Z . This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

Free subgroups in k(x 1,... ,x n )(X;σ) and k(x,y)(k;σ)

2019 ◽  
Vol 31 (3) ◽  
pp. 769-777
Author(s):  
Jairo Z. Gonçalves
Keyword(s):  
Normal Subgroup ◽  
Polynomial Ring ◽  
Multiplicative Group ◽  
Free Subgroup ◽  
Laurent Polynomials ◽  
Skew Polynomial Ring ◽  
Ring Of Fractions ◽  
Skew Polynomial ◽  
Free Subgroups ◽  
Mild Restriction

Abstract Let k be a field, let {\mathfrak{A}_{1}} be the k-algebra {k[x_{1}^{\pm 1},\dots,x_{s}^{\pm 1}]} of Laurent polynomials in {x_{1},\dots,x_{s}} , and let {\mathfrak{A}_{2}} be the k-algebra {k[x,y]} of polynomials in the commutative indeterminates x and y. Let {\sigma_{1}} be the monomial k-automorphism of {\mathfrak{A}_{1}} given by {A=(a_{i,j})\in GL_{s}(\mathbb{Z})} and {\sigma_{1}(x_{i})=\prod_{j=1}^{s}x_{j}^{a_{i,j}}} , {1\leq i\leq s} , and let {\sigma_{2}\in{\mathrm{Aut}}_{k}(k[x,y])} . Let {D_{i}} , {1\leq i\leq 2} , be the ring of fractions of the skew polynomial ring {\mathfrak{A}_{i}[X;\sigma_{i}]} , and let {D_{i}^{\bullet}} be its multiplicative group. Under a mild restriction for {D_{1}} , and in general for {D_{2}} , we show that {D_{i}^{\bullet}} , {1\leq i\leq 2} , contains a free subgroup. If {i=1} and {s=2} , we show that a noncentral normal subgroup N of {D_{1}^{\bullet}} contains a free subgroup.

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