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Results tagged with schemes
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user 5101
The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.
14
votes
Building algebraic geometry without prime ideals
This nice approach to points on schemes in fact becomes crucial once one leaves the world of schemes and travels to the galaxy of stacks. …
- 21.4k
7
votes
1
answer
398
views
Higher-dimensional Artin L-functions
I begin by clarifying that the "higher-dimensional" in my question refers to analogues of Artin L-functions over higher dimensional base schemes than $\mathrm{Spec}(\mathbb{Z})$.
Now for the set-up. …
- 21.4k
6
votes
Accepted
Minimal fields of isomorphism for varieties
Yes if $K=\mathbb{R}$ for example, but no in general.
Namely this fails for curves of genus $1$, over $\mathbb{Q}$, say. Given an elliptic curve $E$ over $\mathbb{Q}$ and a positive integer $d$, a ge …
- 21.4k
4
votes
Accepted
Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?
This answer addresses the second question: "If we assume the fibers of $\pi$ are curves of genus $0$, can we embed $X$ into $\mathbb{P}^2_S$?"
The answer to this is also no (providing there is a singu …
- 21.4k
3
votes
Accepted
Conjugate surfaces: informations about the orbits
This problem is of a more arithmetic nature, than geometric.
For example, since every automorphism of $\mathbb{C}$ preserves $\mathbb{Q}$, we see that if $X$ can be defined over $\mathbb{Q}$, then $\ …
- 21.4k
2
votes
Embeddings of fields and rational points
Let $\eta \to S$ be the generic point of $S$. The function field of $S$ is exactly the residue field of $\eta$. Let $k(S) \subset K$ be a field extension (no need to assume that $K$ is algebraically c …
- 21.4k
2
votes
Accepted
Dimension of Zariski closure of a closed point of generic fiber
Probably the easiest way to prove this is via flatness.
The closure $\bar{x}$ is integral and dominates $S$, thus is flat over $S$ (see Proposition III.9.7 in Hartshorne). The dimension of the fibres …
- 21.4k
1
vote
Doing scheme theory with Hausdorff spaces
The replacement in algebraic geometry of the Hausdorff condition is that of a separated scheme. These behave in many respects like Hausdorff spaces, for the Zariski topology.
But despite the Zarski t …
- 21.4k