scholarly journals On hyperquadrics containing projective varieties

2020 ◽  
Vol 32 (5) ◽  
pp. 1199-1209
Author(s):  
Euisung Park
Keyword(s):  
Projective Variety ◽  
Minimal Degree ◽  
Zero Set ◽  
Projective Varieties ◽  
Quadratic Equations ◽  
Linearly Independent ◽  
Del Pezzo ◽  
Classification Of Varieties ◽  
Irreducible Projective Variety

AbstractClassical Castelnuovo Lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension c is at most {{{c+1}\choose{2}}} and the equality is attained if and only if the variety is of minimal degree. Also G. Fano’s generalization of Castelnuovo Lemma implies that the next case occurs if and only if the variety is a del Pezzo variety. Recently, these results are extended to the next case in [E. Park, On hypersurfaces containing projective varieties, Forum Math. 27 2015, 2, 843–875]. This paper is intended to complete the classification of varieties satisfying at least {{{c+1}\choose{2}}-3} linearly independent quadratic equations. Also we investigate the zero set of those quadratic equations and apply our results to projective varieties of degree {\geq 2c+1}.

scholarly journals On hypersurfaces containing projective varieties

2015 ◽  
Vol 27 (2) ◽  
Author(s):  
Euisung Park
Keyword(s):  
Projective Variety ◽  
Projective Varieties ◽  
Quadratic Equations ◽  
Linearly Independent ◽  
Irreducible Projective Variety

AbstractClassical Castelnuovo's lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension

scholarly journals Polarized Rigid Del Pezzo Surfaces in Low Codimension

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1567
Author(s):  
Muhammad Imran Qureshi
Keyword(s):  
Computer Algebra System ◽  
Symmetric Matrix ◽  
Single Equation ◽  
Complete Classification ◽  
Computer Assisted ◽  
Del Pezzo Surfaces ◽  
Linearly Independent ◽  
Del Pezzo ◽  
Detailed Computer

We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type ki×1ri(1,ai):3≤ri≤10,ki∈Z≥0 as well-formed and quasismooth varieties embedded in some weighted projective space. In particular, we present a collection of 147 such surfaces such that their image under their anti-canonical embeddings can be described by using one of the following sets of equations: a single equation, two linearly independent equations, five maximal Pfaffians of 5×5 skew symmetric matrix, and nine 2×2 minors of size 3 square matrix. This is a complete classification of such surfaces under certain carefully chosen bounds on the weights of ambient weighted projective spaces and it is largely based on detailed computer-assisted searches by using the computer algebra system MAGMA.

Varieties with nef diagonal

2019 ◽  
Vol 31 (02) ◽  
pp. 2050011 ◽  
Author(s):  
Taku Suzuki ◽  
Kiwamu Watanabe
Keyword(s):  
Projective Variety ◽  
Smooth Projective Variety ◽  
Complete Intersections ◽  
Spherical Varieties ◽  
Del Pezzo

For a smooth projective variety [Formula: see text], we consider when the diagonal [Formula: see text] is nef as a cycle on [Formula: see text]. In particular, we give a classification of complete intersections and smooth del Pezzo varieties where the diagonal is nef. We also study the nefness of the diagonal for spherical varieties.

scholarly journals Regularity of structure sheaves of varieties with isolated singularities

2020 ◽  
pp. 2050039
Author(s):  
Joaquín Moraga ◽  
Jinhyung Park ◽  
Lei Song
Keyword(s):  
Recent Result ◽  
Projective Variety ◽  
Isolated Singularities ◽  
Projective Varieties ◽  
Multiplier Ideals

Let [Formula: see text] be a non-degenerate normal projective variety of codimension [Formula: see text] and degree [Formula: see text] with isolated [Formula: see text]-Gorenstein singularities. We prove that the Castelnuovo–Mumford regularity [Formula: see text], as predicted by the Eisenbud–Goto regularity conjecture. Such a bound fails for general projective varieties by a recent result of McCullough–Peeva. The main techniques are Noma’s classification of non-degenerate projective varieties and Nadel vanishing for multiplier ideals. We also classify the extremal and the next to extremal cases.

On Projective Varieties with Projectively Equivalent Zero-Dimensional Linear Sections

1992 ◽  
Vol 35 (1) ◽  
pp. 3-13 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Positive Characteristic ◽  
Projective Varieties ◽  
Partial Classification ◽  
Linear Sections ◽  
Classification Of Varieties

AbstractHere we give a partial classification of varieties X ⊂ Pn such that any two general zero-dimensional linear sections are projectively equivalent. They exist (with deg(X) > codim(X) + 2) only in positive characteristic.

WEIGHTED GRIDS IN COMPLEX JORDAN* TRIPLES

2010 ◽  
Vol 03 (01) ◽  
pp. 155-184
Author(s):  
L. L. STACHÓ
Keyword(s):  
Independent Sets ◽  
Vector Spaces ◽  
Complete List ◽  
Real Vector ◽  
Linearly Independent

Weighted grids are linearly independent sets {gw : w ∈ W} of signed tripotents in Jordan* triples indexed by figures W in real vector spaces such that {gugvgw} ∈ ℂgu-v+w (= 0 if u - v + w ∉ W). They arise naturally as systems of weight vectors of certain abelian families of Jordan* derivations. Based on Neher's grid theory, a classification of association free non-nil weighted grids is given. As a first step beyond the setting of classical grids, the complete list of complex weighted grids of pairwise associated signed tripotents indexed by ℤ2 is established.

scholarly journals Codimension One Holomorphic Distributions on the Projective Three-space

2018 ◽  
Vol 2020 (23) ◽  
pp. 9011-9074 ◽  
Author(s):  
Omegar Calvo-Andrade ◽  
Maurício Corrêa ◽  
Marcos Jardim
Keyword(s):  
Projective Space ◽  
Moduli Space ◽  
Tangent Bundle ◽  
Projective Variety ◽  
Arbitrary Degree ◽  
Codimension One ◽  
Quot Scheme ◽  
Space Of Distributions ◽  
Three Space

Abstract We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves.

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