{"docId":8404,"paperId":6792,"url":"https:\/\/epiga.episciences.org\/6792","doi":"10.46298\/epiga.2021.6792","journalName":"\u00c9pijournal de G\u00e9om\u00e9trie Alg\u00e9brique","issn":"","eissn":"2491-6765","volume":[{"vid":429,"name":"Volume 5"}],"section":[],"repositoryName":"arXiv","repositoryIdentifier":"2009.07381","repositoryVersion":2,"repositoryLink":"https:\/\/arxiv.org\/abs\/2009.07381v2","dateSubmitted":"2020-09-22 07:01:41","dateAccepted":"2021-06-23 21:14:00","datePublished":"2021-08-31 09:15:30","titles":["Torus actions, Morse homology, and the Hilbert scheme of points on\n affine space"],"authors":["Totaro, Burt"],"abstracts":["We formulate a conjecture on actions of the multiplicative group in motivic homotopy theory. In short, if the multiplicative group G_m acts on a quasi-projective scheme U such that U is attracted as t approaches 0 in G_m to a closed subset Y in U, then the inclusion from Y to U should be an A^1-homotopy equivalence. We prove several partial results. In particular, over the complex numbers, the inclusion is a homotopy equivalence on complex points. The proofs use an analog of Morse theory for singular varieties. Application: the Hilbert scheme of points on affine n-space is homotopy equivalent to the subspace consisting of schemes supported at the origin.","Comment: 14 pages"],"keywords":["Mathematics - Algebraic Geometry","Mathematics - Algebraic Topology","Mathematics - K-Theory and Homology","14L30 (Primary) 14C05, 14F42, 55R80 (Secondary)"]}