Timeline for Genus of non-reduced curves
Current License: CC BY-SA 4.0
6 events
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| Apr 23, 2019 at 16:21 | comment | added | Giulio | thanks! so here the upshot is that the complete intersection has genus 1 and $B_{red}$ has genus 0, which is exactly what I am looking for, i.e. the genus of $B_{red}$ is less than say something like $1/2$ or $1/4$ the genus of the complete intersection (this is exactly what happens when $B$ is a complete intersection!) | |
| Apr 23, 2019 at 15:44 | comment | added | roy smith | It seems the reduced curves can both have genus zero, but maybe you meant "less than or equal"? I.e. a rank 1 quadric cuts a quadric cone in a double conic, reduced genus zero. Then a rank 2 quadric containing a ruling line, cuts a double line and a residual conic. The double line again has reduced genus zero. These sections of O(2) are not generic, but it seems unlikely that generic sections would meet in a non reduced curve. Is it possible that B generally double covers its reduced form? I.e. choose the second section to contain the reduced component and let it move. | |
| Apr 23, 2019 at 14:39 | comment | added | Giulio | Thank you very much!! My main problem is actually how to relate the genera of $B$ and $B_{red}$ when $B$ is not a complete intersection in $X$. (this seems easy to me just when $B$ itself is a complete intersection). | |
| Apr 23, 2019 at 2:21 | comment | added | roy smith | The same argument seems to give in general 2(p-p') = (C-C').(D+E+K), where D,E are divisors on the 3 fold X, K is the canonical divisor on X, and these intersection numbers are taken on X. Thus even in the projective space case, it seems that to get a good hold on the genus of C', one needs not only the genus of C, but also its degree, i.e. its intersection numbers with K,S,S'. Still it is tempting to think the answer to your question is yes, even with no formulas. Oops, poor notation; my C,C' are your curves A,B. | |
| Apr 22, 2019 at 21:40 | history | edited | Michael Albanese | CC BY-SA 4.0 |
added 141 characters in body
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| Apr 22, 2019 at 21:19 | history | answered | roy smith | CC BY-SA 4.0 |