scholarly journals The Dedekind different of a Cayley–Bacharach scheme

2019 ◽  
Vol 18 (02) ◽  
pp. 1950027 ◽  
Author(s):  
Martin Kreuzer ◽  
Tran N. K. Linh ◽  
Le Ngoc Long
Keyword(s):  
Projective Space ◽  
Algebraic Structure ◽  
Coordinate Ring

Given a 0-dimensional scheme [Formula: see text] in a projective space [Formula: see text] over a field [Formula: see text], we characterize the Cayley–Bacharach property (CBP) of [Formula: see text] in terms of the algebraic structure of the Dedekind different of its homogeneous coordinate ring. Moreover, we characterize Cayley–Bacharach schemes by Dedekind’s formula for the conductor and the complementary module, we study schemes with minimal Dedekind different using the trace of the complementary module, and we prove various results about almost Gorenstein and nearly Gorenstein schemes.

scholarly journals The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Daniele Faenzi ◽  
Joan Pons-Llopis
Keyword(s):  
Projective Space ◽  
Representation Type ◽  
Coordinate Ring

We show that all reduced closed subschemes of projective space that have a Cohen-Macaulay graded coordinate ring are of wild Cohen-Macaulay type, except for a few cases which we completely classify.

On the Canonical Module of A 0-Dimensional Scheme

1994 ◽  
Vol 46 (2) ◽  
pp. 357-379 ◽  
Author(s):  
Martin Kreuzer
Keyword(s):  
Algebraic Structure ◽  
General Position ◽  
Free Resolution ◽  
Higher Order ◽  
Main Topic ◽  
Hilbert Functions ◽  
Coordinate Ring ◽  
Minimal Free Resolution ◽  
Canonical Module ◽  
Geometric Properties

AbstractThe main topic of this paper is to give characterizations of geometric properties of O-dimensional subschemes in terms of the algebraic structure of the canonical module of their projective coordinate ring. We characterize Cayley- Bacharach, (higher order) uniform position, linearly and higher order general position properties, and derive inequalities for the Hilbert functions of such schemes. Finally we relate the structure of the canonical module to properties of the minimal free resolution of X.

scholarly journals Geometry of nonholonomic complexes NGr (1, 5, 5)

2011 ◽  
Vol 52 ◽  
Author(s):  
Kazimieras Navickis
Keyword(s):  
Differential Geometry ◽  
Projective Space ◽  
Algebraic Structure ◽  
Fundamental Object ◽  
Dimensional Projective Space

In this article the differential geometry of nonholonomic complexes in five dimensional projective space is considered. The algebraic structure of the first fundamental object is considered and the geometrcal interpretations of some comitants of the first fundamental object are given.  

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 Gauged double field theory as an L∞ algebra

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Eric Lescano ◽  
Martín Mayo
Keyword(s):  
Field Theory ◽  
Algebraic Structure ◽  
Classical Field ◽  
Field Theories ◽  
Lie Derivative ◽  
Linear Perturbation ◽  
Double Field Theory ◽  
Generalized Frame ◽  
Symmetry Transformations ◽  
Classical Field Theories

Abstract L∞ algebras describe the underlying algebraic structure of many consistent classical field theories. In this work we analyze the algebraic structure of Gauged Double Field Theory in the generalized flux formalism. The symmetry transformations consist of a generalized deformed Lie derivative and double Lorentz transformations. We obtain all the non-trivial products in a closed form considering a generalized Kerr-Schild ansatz for the generalized frame and we include a linear perturbation for the generalized dilaton. The off-shell structure can be cast in an L3 algebra and when one considers dynamics the former is exactly promoted to an L4 algebra. The present computations show the fully algebraic structure of the fundamental charged heterotic string and the $$ {L}_3^{\mathrm{gauge}} $$ L 3 gauge structure of (Bosonic) Enhanced Double Field Theory.

scholarly journals Polymorphic Iterable Sequential Effect Systems

2021 ◽  
Vol 43 (1) ◽  
pp. 1-79
Author(s):  
Colin S. Gordon
Keyword(s):  
Algebraic Structure ◽  
Position Effect ◽  
Type Systems ◽  
Sequential Effect ◽  
Prior Work ◽  
Sequential Effects ◽  
Wide Range ◽  
The Relationship ◽  
Categorical Semantics

Effect systems are lightweight extensions to type systems that can verify a wide range of important properties with modest developer burden. But our general understanding of effect systems is limited primarily to systems where the order of effects is irrelevant. Understanding such systems in terms of a semilattice of effects grounds understanding of the essential issues and provides guidance when designing new effect systems. By contrast, sequential effect systems—where the order of effects is important—lack an established algebraic structure on effects. We present an abstract polymorphic effect system parameterized by an effect quantale—an algebraic structure with well-defined properties that can model the effects of a range of existing sequential effect systems. We define effect quantales, derive useful properties, and show how they cleanly model a variety of known sequential effect systems. We show that for most effect quantales, there is an induced notion of iterating a sequential effect; that for systems we consider the derived iteration agrees with the manually designed iteration operators in prior work; and that this induced notion of iteration is as precise as possible when defined. We also position effect quantales with respect to work on categorical semantics for sequential effect systems, clarifying the distinctions between these systems and our own in the course of giving a thorough survey of these frameworks. Our derived iteration construct should generalize to these semantic structures, addressing limitations of that work. Finally, we consider the relationship between sequential effects and Kleene Algebras, where the latter may be used as instances of the former.

scholarly journals Embedding Feynman integral (Calabi-Yau) geometries in weighted projective space

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jacob L. Bourjaily ◽  
Andrew J. McLeod ◽  
Cristian Vergu ◽  
Matthias Volk ◽  
Matt von Hippel ◽  
...  
Keyword(s):  
Projective Space ◽  
Feynman Integral ◽  
Weighted Projective Space

scholarly journals Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

2021 ◽  
Vol 9 ◽  
Author(s):  
Alex Chirvasitu ◽  
Ryo Kanda ◽  
S. Paul Smith
Keyword(s):  
Elliptic Curve ◽  
Symmetric Power ◽  
Canonical Homomorphism ◽  
Coordinate Ring ◽  
Characteristic Variety ◽  
Closed Subvariety ◽  
Coordinate Rings ◽  
Cubic Hypersurfaces ◽  
Elliptic Algebras ◽  
Twisted Homogeneous Coordinate Ring

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

Branched Coverings and Minimal Free Resolution for Infinite-Dimensional Complex Spaces

2003 ◽  
Vol 10 (1) ◽  
pp. 37-43
Author(s):  
E. Ballico
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Free Resolution ◽  
Complete Intersections ◽  
Vanishing Theorem ◽  
Sheaf Cohomology ◽  
Minimal Free Resolution ◽  
Branched Coverings ◽  
Infinite Dimensional ◽  
Complex Spaces

Abstract We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1. In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P(V ).

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