scholarly journals Collections of Hypersurfaces Containing a Curve

2018 ◽  
Vol 2020 (13) ◽  
pp. 3927-3977
Author(s):  
Dennis Tseng
Keyword(s):  
Irreducible Component ◽  
Positive Characteristic ◽  
Large Range ◽  
Maximal Dimension ◽  
Higher Dimensional ◽  
Singular Loci

Abstract We consider the closed locus parameterizing $k$-tuples of hypersurfaces that have positive dimensional intersection and fail to intersect properly, and show in a large range of degrees that its unique irreducible component of maximal dimension consists of tuples of hypersurfaces whose intersection contains a line. We then apply our methods in conjunction with a known reduction to positive characteristic argument to find the unique component of maximal dimension of the locus of hypersurfaces with positive dimensional singular loci. We will also find the components of maximal dimension of the locus of smooth hypersurfaces with a higher dimensional family of lines through a point than expected.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

Centers and Azumaya loci for finite W-algebras in positive characteristic

2021 ◽  
pp. 1-32
Author(s):  
BIN SHU ◽  
YANG ZENG
Keyword(s):  
Lie Algebra ◽  
Nilpotent Element ◽  
Positive Characteristic ◽  
Irreducible Representations ◽  
Maximal Dimension ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Maximal Spectrum ◽  
Smooth Locus

Abstract In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

HIGHER-DIMENSIONAL OPERATORS TO THE RESCUE OF MINIMAL SU(5)

1993 ◽  
Vol 08 (16) ◽  
pp. 1487-1494 ◽  
Author(s):  
BISWAJOY BRAHMACHARI ◽  
P.K. PATRA ◽  
UTPAL SARKAR ◽  
K. SRIDHAR
Keyword(s):  
Quantum Gravity ◽  
Extra Dimensions ◽  
Large Range ◽  
Spontaneous Compactification ◽  
Higher Dimensional ◽  
Gravity Effects ◽  
Proton Lifetime ◽  
Range Of Values ◽  
Kaluza Klein

We consider the modification of the minimal SU(5) Lagrangian due to higher-dimensional operators, arising from quantum gravity effects or from spontaneous compactification of extra dimensions in Kaluza-Klein type theories. Due to these operators the SU (3)c, SU (2)L and U (1)Y couplings do not meet at all at the unification scale, MU, and the magnitudes of the mismatch are directly related to the couplings of the higher-dimensional operators. In particular, we consider five- and six-dimensional operators and show that a large range of values of couplings of these operators are compatible with the latest values of sin 2 θW and as derived from LEP, and also with the experimental constraints on MU coming from proton lifetime.

scholarly journals Rational homology and homotopy of high-dimensional string links

2018 ◽  
Vol 30 (5) ◽  
pp. 1209-1235
Author(s):  
Paul Arnaud Songhafouo Tsopméné ◽  
Victor Turchin
Keyword(s):  
Finite Graph ◽  
High Dimensional ◽  
Homotopy Groups ◽  
Maximal Dimension ◽  
Stable Range ◽  
Rational Homotopy ◽  
Rational Homology ◽  
Graph Complexes ◽  
Higher Dimensional ◽  
String Links

AbstractArone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of {\pi_{0}}.

scholarly journals Deformations of rational curves in positive characteristic

2020 ◽  
Vol 2020 (769) ◽  
pp. 55-86
Author(s):  
Kazuhiro Ito ◽  
Tetsushi Ito ◽  
Christian Liedtke
Keyword(s):  
Positive Characteristic ◽  
Rational Curves ◽  
Kodaira Dimension ◽  
Sufficient Criterion ◽  
Higher Dimensional ◽  
Negative Kodaira Dimension

AbstractWe study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than {\frac{1}{2}(p-1)} (resp. p), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher-dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.

scholarly journals GEOMETRIC LOCAL -FACTORS IN HIGHER DIMENSIONS

2021 ◽  
pp. 1-27
Author(s):  
Quentin Guignard
Keyword(s):  
Positive Characteristic ◽  
Higher Dimensions ◽  
Product Formula ◽  
Perfect Field ◽  
Local Factors ◽  
Higher Dimensional ◽  
Vanishing Cycle ◽  
Field Of Positive Characteristic

Abstract We prove a product formula for the determinant of the cohomology of an étale sheaf with $\ell $ -adic coefficients over an arbitrary proper scheme over a perfect field of positive characteristic p distinct from $\ell $ . The local contributions are constructed by iterating vanishing cycle functors as well as certain exact additive functors that can be considered as linearised versions of Artin conductors and local $\varepsilon $ -factors. We provide several applications of our higher dimensional product formula, such as twist formulas for global $\varepsilon $ -factors.

scholarly journals Varieties of a class of elementary subalgebras

2021 ◽  
Vol 7 (2) ◽  
pp. 2084-2101
Author(s):  
Yang Pan ◽  
◽  
Yanyong Hong ◽  
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Positive Characteristic ◽  
Maximal Dimension ◽  
Closed Field ◽  
Characteristic P ◽  
Simple Algebraic Group ◽  
Restricted Lie Algebra ◽  
Irreducible Components ◽  
Type C

<abstract><p>Let $ G $ be a connected standard simple algebraic group of type $ C $ or $ D $ over an algebraically closed field $ \Bbbk $ of positive characteristic $ p &gt; 0 $, and $ \mathfrak{g}: = \mathrm{Lie}(G) $ be the Lie algebra of $ G $. Motivated by the variety of $ \mathbb{E}(r, \mathfrak{g}) $ of $ r $-dimensional elementary subalgebras of a restricted Lie algebra $ \mathfrak{g} $, in this paper we characterize the irreducible components of $ \mathbb{E}(\mathrm{rk}_{p}(\mathfrak{g})-1, \mathfrak{g}) $ where the $ p $-rank $ \mathrm{rk}_{p}(\mathfrak{g}) $ is defined to be the maximal dimension of an elementary subalgebra of $ \mathfrak{g} $.</p></abstract>

Small Computer System for Micrograph Analysis

1981 ◽  
Vol 39 ◽  
pp. 270-271
Author(s):  
J. F. Hainfeld ◽  
J. S. Wall
Keyword(s):  
Large Range ◽  
Small Computer ◽  
Black And White ◽  
Contour Maps ◽  
Points Of Interest ◽  
Computer Aided Analysis ◽  
Specialized Hardware ◽  
Additional Memory ◽  
Selection Of ◽  
Publication Quality

Cost reduction and availability of specialized hardware for image processing have made it reasonable to purchase a stand-alone interactive work station for computer aided analysis of micrographs. Some features of such a system are: 1) Ease of selection of points of interest on the micrograph. A cursor can be quickly positioned and coordinates entered with a switch. 2) The image can be nondestructively zoomed to a higher magnification for closer examination and roaming (panning) can be done around the picture. 3) Contrast and brightness of the picture can be varied over a very large range by changing the display look-up tables. 4) Marking items of interest can be done by drawing circles, vectors or alphanumerics on an additional memory plane so that the picture data remains intact. 5) Color pictures can easily be produced. Since the human eye can detect many more colors than gray levels, often a color encoded micrograph reveals many features not readily apparent with a black and white display. Colors can be used to construct contour maps of objects of interest. 6) Publication quality prints can easily be produced by taking pictures with a standard camera of the T.V. monitor screen.

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