Abelian surfaces in projective toric 4-folds (Q818722)

It is worth while to try to embed abelian surfaces in toric 4-folds. One reason why is that such an embedding can give interesting, usually very exceptional, rank 2 vector bundles (the Horrocks-Mumford bundle is the most famous example). One attempt is by the reviewer [Math. Ann. 313, No. 3, 409--427 (1999; Zbl 0919.14009)]: as the author here politely points out, although the theoretical part of that paper is correct, the example given there is wrong. On a more positive note, the present paper contains three useful results. It is shown that an abelian variety cannot be a hypersurface in a smooth toric variety, and that an abelian surface can only be embedded in the product of a curve and a toric surface in an essentially trivial way. Then, for one particular toric Fano 4-fold (the pseudo del Pezzo 4-fold, \(\widetilde V_4\) in the list of toric Fano 4-folds given by \textit{V. V. Batyrev} [J. Math. Sci., New York 94, No. 1, 1021--1050 (1999; Zbl 0929.14024)] and \textit{H. Sato} [Tohoku Math. J., II. Ser. 52, No. 3, 383--413 (2000; Zbl 1028.14015)]) the author shows that it does contain abelian surfaces, but they are all in a weak sense degenerate (they miss some toric divisor).