scholarly journals Full-Rank Valuations and Toric Initial Ideals

2020 ◽  
Author(s):  
Lara Bossinger
Keyword(s):  
Projective Variety ◽  
Full Rank ◽  
Projective Spaces ◽  
Weight Vector ◽  
Flag Variety ◽  
Coordinate Ring ◽  
Semi Group ◽  
Initial Ideal ◽  
Initial Ideals ◽  
Plabic Graphs

Abstract Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces, and let $A$ be its (multi-)homogeneous coordinate ring. To a full-rank valuation ${\mathfrak{v}}$ on $A$ we associate a weight vector $w_{\mathfrak{v}}$. Our main result is that the value semi-group of ${\mathfrak{v}}$ is generated by the images of the generators of $A$ if and only if the initial ideal of $I$ with respect to $w_{\mathfrak{v}}$ is prime. As application, we prove a conjecture by [ 7] connecting the Minkowski property of string polytopes to the tropical flag variety. For Rietsch-Williams’ valuation for Grassmannians, we identify a class of plabic graphs with non-integral associated Newton–Okounkov polytope (for ${\operatorname *{Gr}}_k(\mathbb C^n)$ with $n\ge 6$ and $k\ge 3$).

scholarly journals Rational Complexity-One $\boldsymbol{T}$-Varieties Are Well-Poised

2017 ◽  
Vol 2019 (13) ◽  
pp. 4198-4232 ◽  
Author(s):  
Nathan Ilten ◽  
Christopher Manon
Keyword(s):  
Prime Ideal ◽  
Affine Space ◽  
Full Rank ◽  
Torus Action ◽  
Coordinate Ring ◽  
Initial Ideal

Abstract Given an affine rational complexity-one $T$-variety $X$, we construct an explicit embedding of $X$ in affine space ${\mathbb{A}}^n$. We show that this embedding is well-poised, that is, every initial ideal of $I_X$ is a prime ideal, and we determine the tropicalization ${\mathrm{Trop}}(X^\circ)$. We then study valuations of the coordinate ring $R_X$ of $X$ which respect the torus action, showing that for full rank valuations, the natural generators of $R_X$ form a Khovanskii basis. This allows us to determine Newton–Okounkov bodies of rational projective complexity-one $T$-varieties, partially recovering (and generalizing) results of Petersen. We apply our results to describe all integral special fibres of ${\mathbb{K}}^*\times T$-equivariant degenerations of rational projective complexity-one $T$-varieties, generalizing a result of Süß and Ilten.

scholarly journals On pliability of del Pezzo fibrations and Cox rings

2017 ◽  
Vol 2017 (723) ◽  
Author(s):  
Hamid Ahmadinezhad
Keyword(s):  
Moduli Space ◽  
Toric Varieties ◽  
Projective Spaces ◽  
Low Rank ◽  
Del Pezzo Surfaces ◽  
Coordinate Ring ◽  
Birational Transformations ◽  
Cox Rings ◽  
Del Pezzo

AbstractWe develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne–Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over

A remark on bigness of the tangent bundle of a smooth projective variety and D-simplicity of its section rings

2015 ◽  
Vol 14 (07) ◽  
pp. 1550098 ◽  
Author(s):  
Jen-Chieh Hsiao
Keyword(s):  
Toric Variety ◽  
Tangent Bundle ◽  
Projective Variety ◽  
Differential Operators ◽  
Smooth Projective Variety ◽  
Flag Variety ◽  
Rings Of Differential Operators

We point out a connection between bigness of the tangent bundle of a smooth projective variety X over ℂ and simplicity of the section rings of X as modules over their rings of differential operators. As a consequence, we see that the tangent bundle of a smooth projective toric variety or a (partial) flag variety is big. Some other applications and related questions are discussed.

scholarly journals ADHM CONSTRUCTION OF PERVERSE INSTANTON SHEAVES

2014 ◽  
Vol 57 (2) ◽  
pp. 285-321 ◽  
Author(s):  
ABDELMOUBINE AMAR HENNI ◽  
MARCOS JARDIM ◽  
RENATO VIDAL MARTINS
Keyword(s):  
Moduli Space ◽  
Projective Variety ◽  
Projective Spaces ◽  
The One ◽  
Fixed Line ◽  
Torsion Free

AbstractWe present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalises the one on projective spaces. This is done by generalising the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterisation of these sheaves in the particular case of projective spaces.

scholarly journals Initial Ideals of Tangent Cones to the Richardson Varieties in the Orthogonal Grassmannian

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Shyamashree Upadhyay
Keyword(s):  
Fixed Point ◽  
Tangent Cone ◽  
Schubert Variety ◽  
Explicit Description ◽  
Tangent Cones ◽  
Initial Ideal ◽  
Term Order ◽  
Initial Ideals ◽  
Orthogonal Grassmannian ◽  
The Ideal

A Richardson variety in the Orthogonal Grassmannian is defined to be the intersection of a Schubert variety in the Orthogonal Grassmannian and an opposite Schubert variety therein. We give an explicit description of the initial ideal (with respect to certain conveniently chosen term order) for the ideal of the tangent cone at any T-fixed point of , thus generalizing a result of Raghavan and Upadhyay (2009). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the Orthogonal-bounded-RSK (OBRSK).

scholarly journals Koszul property for points in projective spaces

2001 ◽  
Vol 89 (2) ◽  
pp. 201 ◽  
Author(s):  
Aldo Conca ◽  
Ngô Viêt Trung ◽  
Giuseppe Valla
Keyword(s):  
Projective Space ◽  
General Position ◽  
Complex Projective Space ◽  
Projective Spaces ◽  
Free Resolution ◽  
General Linear ◽  
Coordinate Ring ◽  
Koszul Property ◽  
Complex Projective

A graded $K$-algebra $R$ is said to be Koszul if the minimal $R$-free graded resolution of $K$ is linear. In this paper we study the Koszul property of the homogeneous coordinate ring $R$ of a set of $s$ points in the complex projective space $\boldsymbol P^n$. Kempf proved that $R$ is Koszul if $s\leq 2n$ and the points are in general linear position. If the coordinates of the points are algebraically independent over $\boldsymbol Q$, then we prove that $R$ is Koszul if and only if $s\le 1 +n + n^2/4$. If $s\le 2n$ and the points are in linear general position, then we show that there exists a system of coordinates $x_0,\dots,x_n$ of $\boldsymbol P^n$ such that all the ideals $(x_0,x_1,\dots,x_i)$ with $0\le i \le n$ have a linear $R$-free resolution.

scholarly journals Geometric collections and Castelnuovo–Mumford regularity

2007 ◽  
Vol 143 (3) ◽  
pp. 557-578 ◽  
Author(s):  
L. COSTA ◽  
R. M. MIRÓ–ROIG
Keyword(s):  
Projective Variety ◽  
Projective Spaces ◽  
Smooth Projective Variety ◽  
Coherent Sheaf ◽  
Projective Varieties ◽  
Coherent Sheaves ◽  
Quadric Hypersurface ◽  
Basic Facts ◽  
Formal Properties ◽  
Smooth Projective Varieties

AbstractThe paper begins by overviewing the basic facts on geometric exceptional collections. Then we derive, for any coherent sheaf $\cF$ on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to $\cF$ and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo–Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties X with a geometric collection σ. We define the notion of regularity of a coherent sheaf $\cF$ on X with respect to σ. We show that the basic formal properties of the Castelnuovo–Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on $\PP^n$ and for a suitable geometric collection of coherent sheaves on $\PP^n$ both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface $Q_n \subset \PP^{n+1}$ (n odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo–Mumford regularity of their extension by zero in $\PP^{n+1}$.

scholarly journals The total coordinate ring of a normal projective variety

2004 ◽  
Vol 276 (2) ◽  
pp. 625-637 ◽  
Author(s):  
E.Javier Elizondo ◽  
Kazuhiko Kurano ◽  
Kei-ichi Watanabe
Keyword(s):  
Projective Variety ◽  
Coordinate Ring
Sign in / Sign up

Export Citation Format

Share Document