Asymptotically maximal families of hypersurfaces in toric varieties

Details Asymptotically maximal families of hypersurfaces in toric varieties Asymptotically maximal families of hypersurfaces in toric varieties

2005-04-23

arxiv.org/abs/math/0504482v1

Mathematics

Algebraic Geometry

18 pages, 1 figure

Scientific paper

A real algebraic variety is maximal (with respect to the Smith-Thom inequality) if the sum of the Betti numbers (with $\mathbb{Z}_2$ coefficients) of the real part of the variety is equal to the sum of Betti numbers of its complex part. We prove that there exist polytopes that are not Newton polytopes of any maximal hypersurface in the corresponding toric variety. On the other hand we show that for any polytope $\Delta$ there are families of hypersurfaces with the Newton polytopes $(\lambda\Delta)_{\lambda \in \mathbb{N}}$ that are asymptotically maximal when $\lambda$ tends to infinity. We also show that these results generalize to complete intersections.

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