Affine Algebraic Varieties
| dc.creator | Zhang, Jing | |
| dc.date | 2007-12-06 | |
| dc.date.accessioned | 2026-07-07T08:47:40Z | |
| dc.date.available | 2026-07-07T08:47:40Z | |
| dc.description | In this paper, we give new criteria for affineness of a variety defined over $\Bbb{C}$. Our main result is that an irreducible algebraic variety $Y$ (may be singular) of dimension $d$ ($d\geq 1$) defined over $\Bbb{C}$ is an affine variety if and only if $Y$ contains no complete curves, $H^i(Y, {\mathcal{O}}_Y)=0$ for all $i>0$ and the boundary $X-Y$ is support of a big divisor, where $X$ is a projective variety containing $Y$. We construct three examples to show that a variety is not affine if it only satisfies two conditions among these three conditions. We also give examples to demonstrate the difference between the behavior of the boundary divisor $D$ and the affineness of $Y$. If $Y$ is an affine variety, then the ring $Γ(Y, {\mathcal{O}}_Y)$ is noetherian. However, to prove that $Y$ is an affine variety, we do not start from this ring. We explain why we do not need to check the noetherian property of the ring $Γ(Y, {\mathcal{O}}_Y)$ directly but use the techniques of sheaf and cohomology. | |
| dc.description | Welcome comments | |
| dc.identifier | https://arxiv.org/abs/0712.0956 | |
| dc.identifier | http://arxiv.org/abs/0712.0956 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/143683 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Complex Variables | |
| dc.subject | 14J10, 14J30, 32E10 | |
| dc.title | Affine Algebraic Varieties | |
| dc.type | text |