New paper: Varieties of general type with small volume

Chengxi Wang and I posted a new paper on the arXiv.

By Hacon-McKernan, Takayama, and Tsuji, there is a constant r_n such that for every r at least r_n, the r-canonical map of every n-dimensional variety of general type is birational. We give examples to show that r_n must grow faster than any polynomial in n.

Related to this, we exhibit varieties of several types (Fano, Calabi-Yau, or general type) with small volume in high dimensions. In particular, we construct a mildly singular (klt) n-fold with ample canonical class whose volume is less than 1/2^(2^n). The klt examples should be close to optimal.

All our examples come from weighted projective hypersurfaces. These exhibit a huge range of behavior, and it is not at all clear how to find the best weighted hypersurfaces for these problems. It’s an attractive problem to explore in combinatorial number theory.

For example, Gavin Brown and Alexander Kasprzyk’s computer program shows that the smallest volume for a weighted hypersurface of dimension 2 which is quasi-smooth (hence klt) and has ample canonical class is 2/57035, about 3.5 x 10^{-5}. This occurs for a general hypersurface of degree 316 in the weighted projective space P(158,85,61,11). What is the pattern behind these numbers? Chengxi Wang and I found one pattern and used it to produce examples with small volume in all dimensions. But one can try to do better.

Image: Maru attempting to minimize his volume.

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