scholarly journals What is the degree of a smooth hypersurface?

2021 ◽  
Vol 23 ◽  
Author(s):  
Antonio Lerario ◽  
Michele Stecconi
Keyword(s):  
Smooth Hypersurface

scholarly journals Codimension bounds for the Noether–Lefschetz components for toric varieties

2021 ◽  
Author(s):  
Ugo Bruzzo ◽  
William D. Montoya
Keyword(s):  
Irreducible Component ◽  
Toric Variety ◽  
Toric Varieties ◽  
Ample Divisor ◽  
Smooth Hypersurface

AbstractFor a quasi-smooth hypersurface X in a projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$ P Σ , the morphism $$i^*:H^p(\mathbb {P}_{\Sigma })\rightarrow H^p(X)$$ i ∗ : H p ( P Σ ) → H p ( X ) induced by the inclusion is injective for $$p=\dim X$$ p = dim X and an isomorphism for $$p<\dim X-1$$ p < dim X - 1 . This allows one to define the Noether–Lefschetz locus $$\mathrm{NL}_{\beta }$$ NL β as the locus of quasi-smooth hypersurfaces of degree $$\beta $$ β such that $$i^*$$ i ∗ acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if $$\dim \mathbb {P}_{\Sigma }=2k+1$$ dim P Σ = 2 k + 1 and $$k\beta -\beta _0=n\eta $$ k β - β 0 = n η , $$n\in \mathbb {N}$$ n ∈ N , where $$\eta $$ η is the class of a 0-regular ample divisor, and $$\beta _0$$ β 0 is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree $$\beta $$ β satisfies the bounds $$n+1\leqslant \mathrm{codim}\,Z \leqslant h^{k-1,\,k+1}(X)$$ n + 1 ⩽ codim Z ⩽ h k - 1 , k + 1 ( X ) .

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).

A representation of distributions supported on smooth hypersurfaces of Rn

1995 ◽  
Vol 117 (1) ◽  
pp. 153-160
Author(s):  
Kanghui Guo
Keyword(s):  
Gaussian Curvature ◽  
Dual Space ◽  
Smooth Hypersurface ◽  
Zero Gaussian Curvature ◽  
Schwartz Class ◽  
Image Position

Let S(Rn) be the space of Schwartz class functions. The dual space of S′(Rn), S(Rn), is called the temperate distributions. In this article, we call them distributions. For 1 ≤ p ≤ ∞, let FLp(Rn) = {f:∈ Lp(Rn)}, then we know that FLp(Rn) ⊂ S′(Rn), for 1 ≤ p ≤ ∞. Let U be open and bounded in Rn−1 and let M = {(x, ψ(x));x ∈ U} be a smooth hypersurface of Rn with non-zero Gaussian curvature. It is easy to see that any bounded measure σ on Rn−1 supported in U yields a distribution T in Rn, supported in M, given by the formula

Tangent bundle of hypersurfaces in G/P

2008 ◽  
Vol 4 (1) ◽  
pp. 91-100
Author(s):  
Indranil Biswas ◽  
Georg Schumacher
Keyword(s):  
Line Bundle ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Smooth Hypersurface ◽  
Canonical Line Bundle

AbstractLet G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and hold; here n = dimH, and τ(G/P) ∈ is the index of G/P which is defined by the identity = where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.

Restriction of the Tangent Bundle of G/P to a Hypersurface

2010 ◽  
Vol 53 (2) ◽  
pp. 218-222
Author(s):  
Indranil Biswas
Keyword(s):  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Tangent Bundle ◽  
Linear Algebraic Group ◽  
Smooth Hypersurface

AbstractLet P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over ℂ, such that n := dimℂG/P ≥ 4. Let ι : Z ↪ G/P be a reduced smooth hypersurface of degree at least (n – 1) · degree(T(G/P))/n. We prove that the restriction of the tangent bundle ι*TG/P is semistable.

On the dual and hessian mappings of projective hypersurfaces

1987 ◽  
Vol 101 (3) ◽  
pp. 461-468 ◽  
Author(s):  
A. D. R. Choudary ◽  
A. Dimca
Keyword(s):  
Projective Space ◽  
Complex Projective Space ◽  
First Order ◽  
Smooth Hypersurface ◽  
Numerical Invariants ◽  
Dual Mapping ◽  
Complex Projective

We investigate the first-order Thom–Boardman singularity sets of the dual mapping for an arbitrary (and then for a generic) smooth hypersurface in the complex projective space ℙn. Our results focus on nonemptiness, connectedness, regular stratifications and numerical invariants for these sets.

SUFFICIENT CONDITIONS FOR INTERPOLATION AND SAMPLING HYPERSURFACES IN THE BERGMAN BALL

2007 ◽  
Vol 18 (05) ◽  
pp. 559-584 ◽  
Author(s):  
TAMÁS FORGÁCS ◽  
DROR VAROLIN
Keyword(s):  
Recent Work ◽  
Fock Space ◽  
Sufficient Conditions ◽  
Higher Dimensions ◽  
Sampling Theorem ◽  
Interpolation Theorem ◽  
Smooth Hypersurface ◽  
Present Setting ◽  
Geometric Density ◽  
Natural Method

We give sufficient conditions for a closed smooth hypersurface W in the n-dimensional Bergman ball to be interpolating or sampling. As in the recent work [5] of Ortega-Cerdà, Schuster and the second author on the Bargmann–Fock space, our sufficient conditions are expressed in terms of a geometric density of the hypersurface that, though less natural, is shown to be equivalent to Bergman ball analogs of the Beurling-type densities used in [5]. In the interpolation theorem we interpolate L2 data from W to the ball using the method of Ohsawa–Takegoshi, extended to the present setting, rather than the Cousin I approach used in [5]. In the sampling theorem, our proof is completely different from [5]. We adapt the more natural method of Berndtsson and Ortega-Cerdà [1] to higher dimensions. This adaptation motivated the notion of density that we introduced. The approaches of [5] and the present paper both work in either the case of the Bergman ball or of the Bargmann–Fock space.

scholarly journals On the order of an automorphism of a smooth hypersurface

2013 ◽  
Vol 197 (1) ◽  
pp. 29-49 ◽  
Author(s):  
Víctor González-Aguilera ◽  
Alvaro Liendo
Keyword(s):  
Smooth Hypersurface

Envelopes and characteristics

1986 ◽  
Vol 100 (3) ◽  
pp. 475-492 ◽  
Author(s):  
J. W. Bruce
Keyword(s):  
Local Structure ◽  
Space Curve ◽  
Function Germ ◽  
Smooth Hypersurface ◽  
Canal Surfaces ◽  
Fixed Radius ◽  
Regular Value ◽  
Smooth Map

Envelopes of hypersurfaces arise in many different geometric situations. For example, the canal surfaces associated with a space curve can be obtained as the envelope of spheres of a fixed radius centred on that curve. (Other examples can be found in [11], chapter II and [4], chapter 5). In what follows we shall be entirely concerned with the local structure of such envelopes. So essentially we have a smooth map germ with 0 a regular value of the restriction of F to . For near 0 we have a smooth hypersurface near , where . The envelope of this family of hypersurfaces is obtained by eliminating t from the equations . Moreover, it can be identified with the discriminant of F viewed as an unfolding, by the x variables, of the function germ . This observation, together with standard results on versal unfoldings of functions, allows one to determine the local structure of many ‘generic’ geometric envelopes. See [2] for details.

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