The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. ...

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14
votes
3answers
536 views
+100

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
6
votes
0answers
65 views
+100

Why is there no theory of $G$-ic varieties, for linear algebraic groups $G$?

A toric variety is an algebraic variety $X$ with an embedding $T \hookrightarrow X$ of an algebraic torus $T$ as a dense open set, such that $T$ acts on $X$ and the embedding is equivariant. It ...
1
vote
1answer
62 views
+50

A possible mistake in Hartshorne chapter 2 proposition 2.6

Here is the context of this question. Hartshorne claim that $O_X(U)\cong \beta_*(O_V)(U)=O_V(\beta^{-1}(U))$ for any open $U\subset X=\operatorname{Spec}A$,but it is possible that ...