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Questions tagged [algebraic-geometry]

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. This tag should not be used for elementary problems which involve both algebra and geometry.

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. This tag should not be used for elementary problems which involve both algebra and geometry.

29074 questions
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How are $\operatorname{Spec} \mathbb{Q}, \operatorname{Spec}\mathbb{R}, \operatorname{Spec}\mathbb{C}$ etc different?

By definition $\operatorname{Spec}k$ is a point for any field $k$. So $\operatorname{Spec}\mathbb{Q}, \operatorname{Spec}\mathbb{R}, \operatorname{Spec}\mathbb{C}$ etc are all the same as topological spaces. But according to the natural inclusion…
M. K.
  • 5,021
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Why is a smooth connected scheme irreducible?

Why is a smooth connected scheme (say over a field) necessarily irreducible? Intuitively it makes sense because we might very well expect points in the intersection of two irreducible components to be singular points. But what is a proof? Feel free…
John Cromwell
  • 315
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Exact sequence of sheaves if and only if exact on the stalks

This is a follow up question to something I asked earlier: What does it mean for a sequence of sheaves to be exact Let $F, G, H$ be sheaves on a topological space $X$, and let $$F \xrightarrow{\alpha} G \xrightarrow{\beta} H$$ be morphisms of…
D_S
  • 33,891
17
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automorphism of the projective space $\mathbb{P}_A^n$

In exercise 16.4.B of Vakil's notes, he establishes that the group of automorphisms of $\mathbb{P}_k^n$ is $PGL_{n+1}(k)$. This I can manage to show, but in the remarks following the exercise he asks why this does not work over an arbitrary base…
adrido
  • 2,283
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Locally Free Sheaves

If $X$ is a locally noetherian scheme and $F$ is a coherent sheaf, I want to show the following equivalence: $F$ is locally free iff its stalk is a free $O_{X,p}$-module for every $p$ in $X$. => follows from the definition of locally free. <= is…
Yadavv
  • 173
17
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3 answers

The Picard Group of the Affine line with double origin

Let $X$ be the affine line with double origin over a field $k$. It is the scheme obtained gluing two copies of the affine line $\mathbb{A}^1_k$ along the open sets $U_1 = U_2 =\mathbb{A}^1_k - (x)$, where, with abuse of notation, $(x)$ is the point…
Giovanni De Gaetano
  • 3,889
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Morphisms of $k$-schemes who agree on $\overline{k}$-points.

Let $k$ be a field and $X,Y$ be two finite-type $k$ schemes such that $X$ is geometrically reduced. Let $f,g : X \to Y$ be two morphisms of $k$-schemes such that the induced morphisms : $X(\overline{k}) \to Y(\overline{k})$ are equal. How does one…
TheloniousM
  • 285
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2 answers

Use irreducible fibers to show $X$ is irreducible

Let $\pi:X\rightarrow Y$ be a proper morphism to an irreducible variety and all fibers of $\pi$ are nonempty, irreducible, and of the same dimension. Show $X$ must also be irreducible. Thanks (Any hints would work too)
Gazerun
  • 687
17
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Pushforward of pullback of a sheaf

Are there any reasonable hypotheses on a map $f: X \to Y$ and a sheaf $E$ on $Y$ so that $f_* f^* E \cong E$?
A B
  • 451
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3 answers

Dominant rational maps

The University I go to doesn't have any courses in (classical) Algebraic Geometry so I am trying to learn myself. I am fairly comfortable with the content I have covered so far aside from a so called "easy" results which I fail to understand. So my…
M Davolo
  • 691
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2 answers

Which functor does the projective space represent?

I hope this question isn't too silly. It is certainly fundamental, so the answer is likely contained at least implicitly in most sources out there, but I haven't seen it done this way (that is, in this particular functorial manner) in a way which is…
Eivind Dahl
  • 1,657
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Push forward of line bundle and of the associated divisor

Let $X$ and $Y$ be smooth scheme over a Dedekind domain (or over a field if you prefer). Let $f \colon X \to Y$ be a finite and flat morphism and let $D$ be a divisor on $X$. Since $f$ is finite flat, we have a divisor $f_\ast D$ on $Y$ and moreover…
John
  • 169
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3 answers

Twisted sheaf $\mathcal{F}(n)$.

Let $\mathcal{F}$ be a sheaf on a scheme $X$ and $O_X(k)$ as usual. We define $\mathcal{F}(n) = \mathcal{F} \otimes_{O_X} O_X(n)$, I don't undertand this definition. What is this tensor product? Then, if we can try to find examples, if $n=1$ and…
ArthurStuart
  • 4,932
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Infinite coproduct of affine schemes

Let $(X_i)_{i\in I}$ be a family of affine schemes, where $I$ is an infinite set and $X_i = Spec(A_i)$ for each $i \in I$. Let $X$ be a coproduct of $(X_i)_{i\in I}$ in the category of schemes. Let $\Gamma(X, \mathcal{O}_X)$ be the ring of global…
Makoto Kato
  • 42,602
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1 answer

Chern Class = Degree of Divisor?

Is the first chern class the same as the degree of the Divisor? Say, $C$ is some divisor on $M$, is $c_1(\mathcal O (C)) = \text{deg }C$? And say I have some Divisor $D$ with first chern class $c_1(\mathcal{O}(D)) = k[S]$ where $[S]$ is some class…
Tina
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