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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 can we prove that formal smoothness is a property local on the source?

I have learned from this question that, in spite of the gap in the proof of 17.1.6 (i) in EGA IV, we can still verify that a morphism of schemes is formally smooth locally on the source. But, even assuming the results of Raynaud-Gruson, I did not…
Nuno
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When affine variety is complete?

How to prove that an affine variety $X$is complete only if $\dim X=0$? It is clear that in this case $X$ must be a single point. But I don't known why its dimension should be zero. Could anyone help me? Thanks a lot!
yang
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Schur-functor for sheaves

When one is given a partition $\lambda=(\lambda_1,...,\lambda_r)$ and a locally free sheaf $\mathcal{E}$ on for example a Grassmannian variety one can apply the Schur-functor $\Sigma^{\lambda}(\mathcal{E})$ for some partition $\lambda$. Now take an…
user109227
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Examples that the morphism $X\times_k k' \rightarrow X$ is not closed

Let $k$ be a field. Let $k'$ be an extension field of $k$. Let $X$ be a $k$-scheme of finite type. If $k'$ is algebraic over $k$, the morphism $X\times_k k' \rightarrow X$ is integral. Hence it is closed. Suppose $k'$ is not algebraic over $k$. I…
Makoto Kato
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Separatedness of a scheme of finite type over a field

Let $k$ be a field. Let $\bar k$ be an algebraic closure of $k$. Let $X$ be a $k$-scheme of finite type. Suppose $X\times_k \bar k$ is separated over $\bar k$. Is $X$ separated over $k$? If yes, how do you prove it?
Makoto Kato
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Relation between Supp and sheaves

Let $\mathcal{F}$ be a cohorent sheaf on projective scheme $X$. My question is simple... If $\operatorname{dim}\operatorname{Supp}\mathcal{F}$ is zero, then $\mathcal{F}(n) =\mathcal{F}$ for any integer $n$??
Sang Cheol Lee
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Page 134 and 137 in Hartshorne's Algebraic Geometry

In page 134, proposition 6.6 Hartshorne mentions that type 2 is a point $x \in X $ x $ \mathbb A^1 $ of codimension one, whose image in $X$ is the generic point of $X$. I realized that this point $x$ corresponds to a prime ideal $\mathcal p$ of…
Suhas
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An exercise in Silverman

This is self-learning, not homework. Problem: Let $A, B \in \bar{\mathbb{K}}$. Characterize the values of $A, B$ for which each of the following varieties is singular. In particular, as $(A,B)$ ranges over $\mathbb{A}^2$, the "singular values" lie…
AndrewG
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associated points of locally Noetherian scheme

Let $X$ be a locally Noetherian scheme with finite number of associated points. Let $f$ be a function on $X$, that is $f$ is a global section of the structure sheaf, if $f$ vanishes at one associated point, is $f$ a zerodivisor? If $X$ is an…
Yubin
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An irreducible curve of degree 3 has one singular point

Good morning, i got stuck with these exercises. Let $X$ be an hypersurface of degree 3 and suppose that $X$ has two singular points $P$ and $Q$. Let $L_{PQ}$ the line containing $P$ and $Q$. Show that $L_{PQ}\subset X$. Let $F(x,y,z)$ be an…
miguemate
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Compute $H^1(X,\Bbb{Z}_U)$

Let $X = \mathbb{A}^1_k$ with $k$ infinite and $U = X - \{P,Q\}$ and $\mathbb{Z}_U= i_{!}(\mathbb{Z}|_U)$, $\Bbb{Z}$ the constant sheaf. I want to say that $H^1(X,\mathbb{Z}_U) \neq 0$. If it is zero we see exact sequence $$0 \to H^0(X,\mathbb{Z}_U)…
Dylan B.
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Hartshorne Theorem III.5.2 (finite generation of cohomology for coherent sheaves on projective schemes over a noetherian ring)

Hartshorne, Algebraic Geometry, Theorem III.5.2, reads (in part) Theorem 5.2 Let $X$ be a projective scheme over a Noetherian ring, and let $\mathcal{O}_X(1)$ be a very ample invertible sheaf on $X$ over $\operatorname{Spec} A$. Let $\mathscr{F}$…
Daniel McLaury
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linear series vs. linear system on algebraic curves

Could someone please tell me if there is any difference between the concepts "linear series" and "linear systems" on algebraic curves? Also, for smooth plane curves of degree $n$, what is the main difference between linear system of "curves of…
user108555
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Question about sheaves on projective varieties.

I am new to algebraic geometry, and really can't get idea of this: For any product $X_{1} \times X_{2}$ of a projective varieties, with projections $p:X_{1}\times X_{2} \rightarrow X_{1}, q:X_{1}\times X_{2}\rightarrow X_{2}$ and let $L_{1},L_{2}$…
user27759
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a counterexample for the morphism of sheaves

Suppose $\textbf{F}$ and $\textbf{G}$ are two presheaves over a topological space $X$,and $\mu:\textbf{F}\longrightarrow \textbf{G}$ is a morphism of presheaves which is surjective.We have a naturally induced morphism of the associated sheaves…
user14242
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