On Resolving Singularities
| dc.creator | Moody, John Atwell | |
| dc.date | 2001-02-13 | |
| dc.date.accessioned | 2026-07-07T04:40:08Z | |
| dc.date.available | 2026-07-07T04:40:08Z | |
| dc.description | Let V be an irreducible affine algebraic variety over a field k of characteristic zero, and let (f_0,...,f_m) be a sequence of elements of the coordinate ring. There is probably no elementary condition on the f_i and their derivatives which determines whether the blowup of V along (f_0,...,f_m) is nonsingular. The result is that there indeed is such an elementary condition, involving the first and second derivatives of the $f_i,$ provided we admit certain singular blowups, all of which can be resolved by an additional Nash blowup. There is is a particular explicit sequence of ideals R=J_0, J_1, J_2,... \subset R so that V_i=Bl_{J_i}V is the i'th Nash blowup of V, with J_i|J_{i+1} for all i. Applying our earlier paper, V_i is nonsingular if and only if the ideal class of J_{i+1} divides some power of the ideal class of J_i. The present paper brings things down to earth considerably: such a divisibility of ideal classes implies that for some N\ge r+2 J_i^{N-r-2}J_{i+1}^{r+3}=J_i^NJ_{i+2}. Yet note that this identity in turn implies J_{i+2} is a divisor of some power of J_{i+1}. Thus although $V_i$ may fail to be nonsingular, when the identity holds the {\it next} variety V_{i+1} must be nonsingular. Thus the Nash question is equivalent to the assertion that the identity above holds for some sufficiently large i and N. | |
| dc.identifier | https://arxiv.org/abs/math/0102100 | |
| dc.identifier | http://arxiv.org/abs/math/0102100 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/60934 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14a05;32c45 | |
| dc.title | On Resolving Singularities | |
| dc.type | text |