Toric Fano manifolds with large Picard number

Benjamin Assarf, Benjamin Nill

Published 2014-09-25Version 1

Casagrande showed that any toric Fano $d$-fold has Picard number at most $2d$. The equality case is only attained by products of $S_3$, where $S_3$ denotes the projective plane blown up in three torus-invariant points. Toric Fano $d$-folds with Picard number equal to $2d-1$ or $2d-2$ have been completely classified in every dimension. In this paper, we show that for any fixed $k$ there is only a finite number of isomorphism classes of toric Fano $d$-folds $X$ (for arbitrary $d$) with Picard number $2d-k$ such that $X$ is not a product of $S_3$ and a lower-dimensional toric Fano manifold. This verifies the qualitative part of a conjecture in a recent paper by the first author, Joswig, and Paffenholz.