Timeline for Irreducible variety
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
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| May 18, 2016 at 15:49 | comment | added | Jérémy Blanc | For the first point, this is just because $A$ is of dimension $1$ (as it is given by two equations in $\mathbb{P}^3$, with no common factor, check it by looking at the polynomials in $t$), so if it is irreducible and contains a closed line, then it is equal to this line (otherwise you could write it as the union of a closed line and the union of other components of dimension $\le 1$ and it is not irreducible). For the second point, an isomorphism is defined everywhere, by definition. You should probably have a look at some basic books of algebraic geometry and go to math.stackexchange. | |
| May 17, 2016 at 16:37 | comment | added | Vadim | Thanks! Regarding the first point, are there any formal statements in any books or other sources that explicitly say that a line is not a part of an irreducible variety. And regarding the second point, it is fine, as in this geometry maps are defined where they are defined, indicated by the broken arrow "-->", so it should be isomorphism on its domain, but, of course, I believe, it should be one-to-one there. That is the problem. | |
| Apr 23, 2016 at 21:33 | history | answered | Jérémy Blanc | CC BY-SA 3.0 |