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.
31,076 questions
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Chern class of tangent bundle of hypersurface of CP^n viewed as complex of coherent sheaves
I would like to compute the total Chern class of the tangent bundle to a hypersurface $X$ of degree $d$ in $\mathbb{P}^n$ by viewing the following short exact sequence as a complex of coherent sheaves ...
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Construction of Base Change/Comparison Map $g^*f_*F \to f'_* g'^*F$
Let us consider a cartesian diagram of schemes
$$
\require{AMScd}
\begin{CD}
X'=X \times_S S' @>{g'} >> X \\
@VVf'V @VVfV \\
Y' @>{g}>> Y
\end{CD}
$$
and let $F$ a sheaf on $X$.
...
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Quotient of $\mathrm{GL}_2(\mathbb{C})$ by a finite group
Let us consider the algebraic group $G=\mathrm{GL}_2(\mathbb{C})$ and consider the $S_2$-action given by conjugation with $P_0=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, that is, the $S_2$-...
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When do coequalizers of algebraic stacks exist?
This question might need some work to actually get a "good" answer.
Here's the background motivation: the $2$-category of algebraic stacks has fibre products and products and so has ...
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Why elliptic curves instead of Edwards curves?
I have been wondering this for a while; I get how elliptic curves of the Weierstrass form $y^2=4x^3-g_2x-g_3$ have the lowest degree and are the simplest way to study a genus $1$ curve. But why aren’t ...
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Preimage of curves in a double cover of surface
Let $S$ be a smooth complex projective surface. We consider the double cover $\pi:X\to S$ branched along $B\equiv 2L$(suppose $B$ is smooth). Now let $C\subset X$ be an irreducible curve.
If $C$ is ...
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Are there any lines on an Abelian surface?
I'm working in the category of schemes over $\mathbb C$ or algebraic varieties over the same. Here by line I mean any curve isomorphic to $\mathbb P^1$, of any degree; i.e. the twisted cubic is a line....
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When is an open immersion affine as a morphism of schemes?
This might be a stupid question but I could not convince myself of the answer.
Let $j:X \to Y$ be an open immersion of schemes, and assume that $j$ is affine. Very broadly, what can we say? More ...
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Hironaka’s polyhedron.
Hironaka proved the resolution of singularities for varieties over characteristic zero. He invented his original invariant associated to the given singular loci. I remember that after blowing ups, his ...
- 1,067
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Quot scheme of a line on $\Bbb A^3$
Let $L \subset \Bbb A^3$ be a line. I'm trying to compute the dimension of the scheme $\operatorname{Quot}_{\Bbb A^3}(\mathscr{I}_L,2)$ and show that it is singular, but I have a bit of trouble with ...
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When is a zero-dimensional subscheme general or special?
On complex algebraic surfaces, say $X$, I'm working on some cohomology calculations of sheaves of ideals of the form $\mathcal{I}_{Z}(L)$, where $L\in\text{Pic}(X)$ and $Z\subset X$ is a zero-...
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Reduced structure commutes with base change
Let $X$ be a scheme, and let $K$ be a characteristic $0$ field.
This post shows that if $X$ is reduced, then base change over $X$ is also reduced. Is it also true that
$$X_{\mathrm{red}}\times_{\...
- 1,656
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how to show that a family of affine curves is non-singular for a family which produces infinitely many curves [closed]
I am working on a proof, and my strategy is to basically prove that this infinite family has no singularity within itself, and then using Siegel I could just prove that there are finitely many ...
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An algebraic structure with uniquely complemented and non-distributive
Is there a concrete example of an algebraic structure with two binary operations that is commutative, associative, uniquely complemented, and non-distributive?
Here is an explicit example that is easy ...
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A Consequence from Grauert's Result on Cohomology of Fibres
Let $f:X \to Y$ be a map of schemes and $y \in Y$ a point with residue field $\kappa(y)$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then there exist a comparison map
$$ f_*(\mathcal{F}) \otimes \...
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