Timeline for Intersections of hypersurfaces of degree $d$ in $\mathbb CP^n$
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| May 19 at 14:19 | comment | added | LittleBear | @JasonStarr: Sorry for a silly question: why the graded ideal generated by degree 3 polynomials is not $(x^2y,xy^2,x^2z,xz^2,y^2z,yz^2,xyz,xyw,xzw,yzw)$? If so, then it seems that its associated closed subscheme is equal to $X$. | |
| Nov 28, 2014 at 15:56 | comment | added | Jason Starr | Just to make that last point a bit more clear, if one starts with the homogeneous coordinate ring $k[x,y,z,w]$ of $\mathbb{P}^3$, for the degree 3 closed subscheme $X$ with graded ideal $\langle xy, xz, yz \rangle$, the graded ideal generated by the degree $3$ homogeneous polynomials above is $\langle x^2y, xy^2, x^2z, xz^2, y^2z, yz^2 \rangle$, whose associated closed subscheme does not equal $X$ (although they are set-theoretically equal). | |
| Nov 28, 2014 at 15:03 | comment | added | Jason Starr | Just to make one comment: this result is "set-theoretic", not "scheme-theoretic". If $X$ is smooth, the result is scheme-theoretic. (Mumford discusses this in an old paper -- forgot the reference, but Lazarsfeld's "Positivity in Algebraic Geometry" gives the Mumford reference in the discussion of C-M regularity for smooth schemes.) For a reducible counterexample, consider the union of the three coordinate axes in $\mathbb{P}^3$: correct set-theoretic intersection, but wrong scheme structure at the singular point. | |
| Nov 28, 2014 at 11:47 | comment | added | Turbo | Is there a textbook reference for this? | |
| Nov 27, 2014 at 19:46 | vote | accept | aglearner | ||
| Nov 27, 2014 at 19:45 | history | answered | abx | CC BY-SA 3.0 |