Numerical obstructions to abelian surfaces in toric Fano 4-folds

On the list of smooth toric Fano 4-folds ([1, 12]), of which there are 124, there remain 19 for which it is not known whether there are embedded abelian surfaces. In order to avoid trivialities, we consider only totally nondegenerate embedded abelian surfaces, in the sense of [11, 12]. Definition 1 An embedding φ: Y ↩ → X from a smooth variety Y to a smooth toric 4-fold X, or its image φ(Y) ⊂ X, is totally nondegenerate if φ(Y) ∩ D � = ∅ for any torus-invariant prime divisor D. Results from [3, 4, 5, 6, 7, 11, 13] about totally nondegenerate embeddings of abelian surfaces in toric Fano 4-folds may be summarised as follows. Theorem 1 Suppose X is a smooth toric Fano 4-fold. Then X admits a totally nondegenerate abelian surface if X = P 4, if X = P 1 × P 3 (type B4)