Extended Abstract: Computational Tasks of Resolution of Singularities

Challenges in Symbolic Computation Software

计算机科学;物理学

Anne Frühbis-Krüger

Others  :  http://drops.dagstuhl.de/opus/volltexte/2006/771/pdf/06271.FruehbisKruegerAnne.ExtAbstract.771.pdf
PID  :  6601

学科分类：计算机科学（综合）

来源: CEUR

【 摘 要 】

When studying a variety (or scheme) which is singular, many properties may also be obtained from a non-singular variety which does not differ too much from the original variety. More precisely, given a variety X this approach requires a non-singular variety ~ X and a proper birational morphism π : ~ X →X which leaves X\Sing(X) unchanged. In a certain sense we may consider ~ X as a kind of smooth model of our given variety. ~ X (and the process of finding it) is called a resolution of singularities or desingularisation of X. There are many known special cases in which this task can be completed without major difficulties as,for example, the case of toric varieties, where toric blow-ups allow computation by combinatorial methods, or the case of normal surfaces where iterating blowing ups of the singular points and normalization suffices. In the general case, however, this problem has been a central topic in the research of many mathematicians over the last century and is up to now only solved in characteristic zero (on which we focus here), but still open in positive characteristic.[First Paragraph]