Type: Article

Computational aspects of the higher Nash blowup of hypersurfaces

Duarte, Daniel;

Abstract:

In the present paper, some computational aspects of the higher Nash blowup Nashn(X) of an affine hypersurface X over C are studied. To this end, the author works with the higher-order Jacobian matrix of X, Jacn(F), defined by means of higher-order differential operators acting on the defining equation F of X. A criterion for non-singularity in terms of the rank of the matrix Jacn(F) is proven in Theorem 2.1 (see also Corollary 2.2), and a way to describe the ideal whose blowup is precisely Nashn(X) is given in Proposition 4.11. Finally, a generalization of Nobile's Theorem is proven for a normal hypersurface in Theorem 4.14: the higher Nash modification of a normal hypersurface X is an isomorphism if and only if X is non-singular.