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Residues in toric varieties

Published online by Cambridge University Press:  04 December 2007

EDUARDO CATTANI ,
DAVID COX  and
ALICIA DICKENSTEIN
EDUARDO CATTANI
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA; e-mail: cattani@math.umass.edu
DAVID COX
Affiliation:
Department of Mathematics and Computer Science, Amherst College, Amherst, MA 01002, USA; e-mail: dac@CS.AMHERST.EDU
ALICIA DICKENSTEIN
Affiliation:
Departamento de Mattemática, F.C.E. y N., Universidad de Buenos Aires, Ciudad Universitaria–Pabellón I, 1428 Buenos Aires, Argentina; e-mail: alidick@dm.uba.ar
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Abstract

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We study residues on a complete toric variety $X$, which are defined in terms of the homogeneous coordinate ring of $X$. We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When $X$ is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent $X$ as a quotient $(Y\setminus\{0\})/C\ast$ such that the toric residue becomes the local residue at 0 in $Y$.

Keywords

toric varietiesresiduestoric residueshomogeneous idealsample divisorsGlobal Transformation Laworbifoldsresidual currents
Type
Research Article
Information
Compositio Mathematica , Volume 108 , Issue 1 , August 1997 , pp. 35 - 76
Copyright
© 1997 Kluwer Academic Publishers