On the dual and hessian mappings of projective hypersurfaces

Choudary, A. D. R.; Dimca, Alexandru

doi:10.1017/s0305004100066834

Cited by 8 publications

(6 citation statements)

References 12 publications

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“…This shows that the (n -k) square matrix (2~,b(x)) is degenerate, which contradicts the assumption that V and W are smooth complete intersections. Another interesting geometric application of this type of argument can be found in our paper [3].…”

Section: T(hp He) C T(hp He) U Shp W Sh Ementioning

confidence: 90%

Hypersurface singularities, codimension two complete intersections and tangency sets

Choudary

Dimca²

1987

Geom Dedicata

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Some codimension 2 projective complete intersections are related to local hypersurface singularities. Necessary and sufficient conditions are given such that the singularities obtained are isolated. Incidentally we prove the finitude of the tangency set of two equidimensional smooth complete intersections with distinct multidegrees.It is a classical idea to relate the study of a complex projective variety V c p" to the study of its affine cone CV, defined by the same homogeneous equations as V in C "+1.When V is a smooth complete intersection, the germ (CV, 0) is an isolated complete intersection singularity and, in this way, many global properties of the variety V are related with the properties of the singularity (CV, 0). See,In this note we present a new link between (a class of) projective varieties and singularities. An avatar of our construction can be found in the table in [1].Let V be a closed subvariety in the complex projective space P" which is not a hypersurface, but whose defining ideal Iv=A = C[xo,...,x,J is generated by two homogeneous polynomials P and Q of degrees p = deg P < deg Q = q. The pair (p, q) is called the type of the variety V and it is clearly well defined.Let (Xv, 0) be the hypersurface singularity at the origin of C "+1 defined by the equation P + Q = 0. First we note that the isomorphism class of the singularity (Xv, 0) does not depend on the choice of the pair of generators (P, Q) for the ideal Iv.Next we give a geometric necessary and sufficient condition on the variety V such that the singularity (Xv, 0) is isolated. It is perhaps more useful and attractive to state this condition in terms of the hypersurfaces He: P = 0 and H~: Q =0 defined by the generators P and Q in p". For an algebraic variety V we denote by SV its singular locus and for two subvarieties V, W c P" of the same dimension we consider the set of tangency points T(V, W) = {x e (V\SV) ~ (W\SW); TxV = Tx W}.We let ,,] denote the closure of a constructible set A c P". THEOREM 1. The hypersurface singularity (Xv, O) is isolated if and only if Geometriae Dedicata 24 (1987), 255-260.

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Section: T(hp He) C T(hp He) U Shp W Sh Ementioning

confidence: 90%

Hypersurface singularities, codimension two complete intersections and tangency sets

Choudary

Dimca²

1987

Geom Dedicata

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“…Let V f ⊂ PV denote the projective hypersurface associated with f . In [4] the authors consider (for k = C) the closed locus S ≥r ⊂ V f where the co-rank of the Hessian H f is ≥ r. They prove that if V f is smooth then for r(r + 1) ≤ dim V , the subvariety S ≥r (V ) is nonempty and codim V f S ≥r (V ) ≤ r(r + 1) 2.…”

Section: Now We Recall Thatmentioning

confidence: 99%

Schmidt rank and singularities

Kazhdan¹,

Polishchuk²

2021

Preprint

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We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous polynomials of degree d (assuming that the characteristic does not divide d(d − 1)). We then use this to relate the Schmidt rank of a homogeneous polynomial (resp., a collection of homogeneous polynomials of the same degree) with the codimension of the singular locus of the corresponding hypersurface (resp., intersection of hypersurfaces).

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“…their closures are disjoint. (i) In the curve case, when n = 2, the equations f = h(f ) = 0 define the inflection points of the curve V (f ), and by Bezout Theorem, their number is at most d(3d − 6), with equality for a generic curve, see for instance [1]. This number is lower for some curves, e.g.…”

Section: Constructible Partitions For the Space Of Homogeneous Polyno...mentioning

confidence: 99%

“…(ii) The subvariety of V (f ) ⊂ P n given by the vanishing of the ideal h n+2−k (f ) corresponds to the closure of the Thom-Boardman singularity set S k (φ f ) of the dual mapping φ f : V (f ) → P n associated to f . For a generic f , the stratum S k (φ f ) is smooth, of codimension k(k+1)/2 in V (f ) (in particular empty if if n−1 < k(k+1)/2) and connected if n − 1 > k(k + 1)/2, see [1].…”

Section: Constructible Partitions For the Space Of Homogeneous Polyno...mentioning

confidence: 99%

Hessian ideals of a homogeneous polynomial and generalized Tjurina algebras

Dimca,

Sticlaru

2014

Preprint

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Using the minors in Hessian matrices, we introduce new graded algebras associated to a homogeneous polynomial. When the associated projective hypersurface has isolated singularities, these algebras are related to some new local algebras associated to isolated hypersurface singularities, which generalize their Tjurina algebras. One consequence of our results is a new way to determine the number of weighted homogeneous singularities of such a hypersurface.

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On the dual and hessian mappings of projective hypersurfaces

Cited by 8 publications

References 12 publications

Hypersurface singularities, codimension two complete intersections and tangency sets

Hypersurface singularities, codimension two complete intersections and tangency sets

Schmidt rank and singularities

Hessian ideals of a homogeneous polynomial and generalized Tjurina algebras

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