 

Alicia Dickenstein,
Sandra Di Rocco, and
Ragni Piene
Interpolation of toric varieties
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print


Published: 
October 5, 2024. 
Keywords: 
Toric variety, interpolation, osculating spaces, lattice polytopes. 
Subject [2010]: 
14M25; 14H45, 14H81, 52B20. 


Abstract
Let X be an mdimensional variety in the ddimensional complex projective space P^{d}. Let k be a positive integer such that the combinatorial number "m+k choose k" is smaller than or equal to d. Consider the following interpolation problem: does there exist a variety Y in P^{d} of dimension strictly smaller than "m+k choose k", with X in Y, such that the tangent space to Y at a point p in X
is equal to the kth osculating space to X at p, for almost all points p in X?
In this paper we consider this question in the toric setting. We prove that if X is toric, then there is a unique toric variety Y solving the above interpolation problem. We
identify Y in the general case and we explicitly compute some of its invariants when X is a toric curve.


Acknowledgements
This work was partially supported by the project Pure Mathematics in Norway, funded by Trond Mohn Foundation and Tromso Research Foundation. AD was partially supported by UBACYT 20020220200166BA and CONICET PIP 20110100580, Argentina. We would like to thank the referees for their valuable comments and questions that have helped us improve the exposition.


Author information
Alicia Dickenstein
Departamento de Matematica, FCEN
Universidad de Buenos Aires
Ciudad Universitaria  Pab. I
(1428) Buenos Aires, Argentina
alidick@dm.uba.ar
Sandra Di Rocco
Department of Mathematics
KTH Royal Institute of Technology
SE 10044 Stockholm, Sweden
dirocco@kth.se
Ragni Piene
Department of Mathematics
University of Oslo
P.O.Box 1053 Blindern, NO0316 Oslo, Norway
ragnip@math.uio.no

