Lesson plan
Rational singularity
In mathematics, more particularly in the field of algebraic geometry, a scheme X has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map : f \colon Y \rightarrow X from a regular scheme Y such that the higher direct images of f_* applied to \mathcal{O}_Y are trivial. That is, : R^i f_* \mathcal{O}_Y = 0 for i > 0 . If there is one such resolution, then it follows that all resolutions share this property, since any two resolutions of singularities can be dominated by a third. For surfaces, rational singularities were defined by .
Article summary
In mathematics, more particularly in the field of algebraic geometry, a scheme X has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a proper birational map : f \colon Y \rightarrow X from a regular scheme Y such that the higher direct images of f_* applied to \mathcal{O}_Y are trivial. That is, : R^i f_* \mathcal{O}_Y = 0 for i > 0 . If there is one such resolution, then it follows that all resolutions share this property, since any two resolutions of singularities can be dominated by a third. For surfaces, rational singularities were defined by .
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.