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Intersection theory on non-commutative surfaces


Author: Peter Jørgensen
Journal: Trans. Amer. Math. Soc. 352 (2000), 5817-5854
MSC (2000): Primary 14A22, 16W50
DOI: https://doi.org/10.1090/S0002-9947-00-02565-4
Published electronically: June 21, 2000
MathSciNet review: 1695026
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Abstract: Consider a non-commutative algebraic surface, $X$, and an effective divisor $Y$ on $X$, as defined by Van den Bergh. We show that the Riemann-Roch theorem, the genus formula, and the self intersection formula from classical algebraic geometry generalize to this setting.

We also apply our theory to some special cases, including the blow up of $X$in a point, and show that the self intersection of the exceptional divisor is $-1$. This is used to give an example of a non-commutative surface with a commutative ${\Bbb P}^1$ which cannot be blown down, because its self intersection is $+1$ rather than $-1$. We also get some results on Hilbert polynomials of modules on $X$.

References [Enhancements On Off] (What's this?)

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Additional Information

Peter Jørgensen
Affiliation: Matematisk Afdeling, Københavns Universitet, Universitetsparken 5, 2100 København Ø, DK-Danmark
Email: popjoerg@math.ku.dk

DOI: https://doi.org/10.1090/S0002-9947-00-02565-4
Keywords: Quasi-scheme, effective divisor, intersection multiplicity, non-commutative surface, non-commutative Riemann-Roch theorem, non-commutative blow up
Received by editor(s): June 16, 1998
Received by editor(s) in revised form: March 31, 1999
Published electronically: June 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society