Type: Article

Higher Nash blowup on normal toric varieties

Duarte, Daniel;

Abstract:

Recently Yasuda has defined higher Nash blow-ups Nashn(X), extending the usual Nash blow-up. In the paper under review the author considers the normalization Nashn(X)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ of Nashn(X) for a toric variety X, defined by a toric ring K[x––a1,…,x––as]. As a first main result he proves that Nashn(X)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ has a structure of toric ring, and its fan can be identified with the Gröbner fan of the ideal Jn:=(x––a1?1,…,x––as?1)n+1 in the ring K[x––a1,…,x––as]; for this purpose the author studies the notion of Gröbner basis of ideals in a toric ring. As a consequence he proves the following basic theorem: If the induced map Nashn(X)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯?X is an isomorphism, then X is nonsingular.