Timeline for Riemann-Roch for Zero Cycles on a Surface

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Oct 22, 2012 at 17:57	comment	added	Jack Huizenga		Yes, for ineffective zero-cycles it is less clear what one should do; however, these cycles are also much less useful in higher dimensions than they are in the curve case. In order to get "new" sections by allowing poles, you must allow poles along a codimension 1 subvariety; allowing poles at points does not give any new sections.
Oct 22, 2012 at 17:03	comment	added	Will Sawin		Thus, one can compute the correct Riemann-Roch formula for $0$-cycles on surfaces the same way one computes the Riemann-Roch formula for $0$-cycles on curves. If $C$ is an effective zero−cycle, $\chi_a(\mathcal O_c)=\operatorname{deg} C$, so $\chi_a(I(C))=\chi_a(\mathcal O_X)-\operatorname{deg} C$. It is not clear to me whether ideal sheaves are well-defined for ineffective zero-cycles.
Oct 22, 2012 at 16:19	history	edited	Jack Huizenga	CC BY-SA 3.0	deleted 18 characters in body
Oct 22, 2012 at 16:12	history	answered	Jack Huizenga	CC BY-SA 3.0