Base locus of linear systems containing a fixed finite set

Abstract We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a homological projective (HP) dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the blowup of $X$ in $X_L$ is naturally HP dual to $Y_L$. The result also holds true when $Y$ is a noncommutative variety or just a category. We extend the result to the case where the base locus $X_L$ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a weakly crepant categorical resolution of singularities of $Y_L$. Finally we give examples where, starting with a noncommutative $Y$, the above process nevertheless gives geometric HP duals.

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Linear Systems Excited by Polynomials of Filtered Poission Pulses

Journal of Applied Mechanics ◽

10.1115/1.2788955 ◽

1997 ◽

Vol 64 (3) ◽

pp. 712-717 ◽

Cited By ~ 15

Author(s):

M. Di Paola

Keyword(s):

Differential Equation ◽

Differential Equations ◽

White Noise ◽

Stochastic Differential Equations ◽

Linear Systems ◽

Moment Equations ◽

Poisson White Noise ◽

Finite Set ◽

The Moment ◽

Memoryless Transformation

The stochastic differential equations for quasi-linear systems excited by parametric non-normal Poisson white noise are derived. Then it is shown that the class of memoryless transformation of filtered non-normal delta correlated process can be reduced, by means of some transformation, to quasi-linear systems. The latter, being excited by parametric excitations, are frst converted into ltoˆ stochastic differential equations, by adding the hierarchy of corrective terms which account for the nonnormality of the input, then by applying the Itoˆ differential rule, the moment equations have been derived. It is shown that the moment equations constitute a linear finite set of differential equation that can be exactly solved.

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Base locus of linear systems on the blowing-up of ℙ3along at most 8 general points

Pacific Journal of Mathematics ◽

10.2140/pjm.2006.223.17 ◽

2006 ◽

Vol 223 (1) ◽

pp. 17-34 ◽

Cited By ~ 1

Author(s):

Cindy De Volder ◽

Antonio Laface

Keyword(s):

Linear Systems ◽

Base Locus ◽

Blowing Up

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Linear Systems on Tropical Curves

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2847 ◽

2010 ◽

Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽

Author(s):

Christian Haase ◽

Gregg Musiker ◽

Josephine Yu

Keyword(s):

Linear System ◽

Linear Systems ◽

Piecewise Linear ◽

Rational Functions ◽

Cell Complex ◽

Linear Functions ◽

Polyhedral Complex ◽

Classical Counterpart ◽

Tropical Curve ◽

Finite Set

International audience A tropical curve $\Gamma$ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system $|D|$ of a divisor $D$ on a tropical curve $\Gamma$ analogously to the classical counterpart. We investigate the structure of $|D|$ as a cell complex and show that linear systems are quotients of tropical modules, finitely generated by vertices of the cell complex. Using a finite set of generators, $|D|$ defines a map from $\Gamma$ to a tropical projective space, and the image can be modified to a tropical curve of degree equal to $\mathrm{deg}(D)$. The tropical convex hull of the image realizes the linear system $|D|$ as a polyhedral complex. Une courbe tropicale $\Gamma$ est un graphe métrique pouvant contenir des arêtes infinies, et une fonction rationnelle tropicale est une fonction continue linéaire par morceaux à pentes entières. Le système linéaire complet $|D|$ d'un diviseur $D$ sur une courbe tropicale $\Gamma$ est défini de façon analogue au cas classique. Nous étudions la structure de $|D|$ en tant que complexe cellulaire et montrons que les systèmes linéaires sont des quotients de modules tropicaux engendrés par un nombre fini de sommets du complexe. Etant donné un ensemble fini de générateurs, $|D|$ définit une application de $\Gamma$ vers un espace projectif tropical, dont l'image peut être modifiée en une courbe tropicale de degré égal à $\mathrm{deg}(D)$. L'enveloppe convexe tropicale de l'image réalise le système linéaire $|D|$ en tant que complexe polyédral.

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