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Symmetric powers do not stabilize


Author: Daniel Litt
Journal: Proc. Amer. Math. Soc. 142 (2014), 4079-4094
MSC (2010): Primary 14C15, 14C25, 14G10
DOI: https://doi.org/10.1090/S0002-9939-2014-12155-1
Published electronically: August 15, 2014
MathSciNet review: 3266979
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Abstract: We discuss the stabilization of symmetric products $ \operatorname {Sym}^n(X)$ of a smooth projective variety $ X$ in the Grothendieck ring of varieties. For smooth projective surfaces $ X$ with non-zero $ h^0(X, \omega _X)$, these products do not stabilize; we conditionally show that they do not stabilize in another related sense, in response to a question of R. Vakil and M. Wood. There are analogies between such stabilization, the Dold-Thom theorem, and the analytic class number formula. Finally, we discuss conjectural Hodge-theoretic obstructions to the stabilization of symmetric products. We provide evidence for these obstructions by showing that the Newton polygon of the motivic zeta function associated to a curve equals the Hodge polygon of the curve.


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Additional Information

Daniel Litt
Affiliation: Department of Mathematics 380-381M, Stanford University, Stanford, California 94305
Email: dlitt@math.stanford.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12155-1
Received by editor(s): October 18, 2012
Received by editor(s) in revised form: January 30, 2013
Published electronically: August 15, 2014
Additional Notes: The author was supported by the NSF GRFP
Communicated by: Lev Borisov
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.