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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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Interior points of Affine Variety in $\mathbb{C}^2$

I've just read that no non-trivial affine variety in $\mathbb{C}^2$ has an interior point. I can't see why this is obviously true. Could somebody give me a hint? In particular, is this something special about $\mathbb{C^2}$, or is it true generally…
Edward Hughes
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Relating a fiber of a sheaf and its cohomology, from Huybrechts & Lehn - The Geom. of Moduli Spaces of Sheaves

Reading the proof of lemma 4.4.4 in Huybrechts and Lehn Geometry of Moduli Spaces of sheaves I come across an isomorphism relating a fibre of an invertible sheaf and its cohomology, and I really don't see where it comes from. Here is the setup, I…
baltazar
  • 1,537
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dimension and intersection

Let $X$ be a scheme (say of finite type over a field), $i: Y \hookrightarrow X$ a closed immersion and $j : U \hookrightarrow X$ be the open complement. Let $F$ be an étale sheaf over $X$ ($\overline{\mathbf{Q}}_l$-sheaf if it helps). In the book…
karénine
  • 71
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How to reduce to the affine case? $\phi(X)$ contains a nonempty open subset of $\overline{\phi(X)}$

Let $\phi: X \rightarrow Y$ be a morphism be varieties over an algebraically closed field. I'm trying to prove that $\phi(X)$ contains a nonempty open subset of $\overline{\phi(X)}$. I know how to solve the problem when $X$ and $Y$ are affine and…
D_S
  • 33,891
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Definition of stalk as a colimit of sheaves

I'm trying to understand the category-theoretic proof that sheafification preserves stalks, using adjoints, as outlined e.g. in this mathoverflow…
WeatherMan
  • 499
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1 answer

Geometric meaning of $\mathcal{F}_x \otimes_{\mathcal{O}_{X,x}} \kappa(x)$

Let $X$ be a noetherian scheme and let $\mathcal{F}$ be a coherent sheaf on $X$. Let $x \in X$ be a point, then what is the geometric meaning of the vector space $\mathcal{F}_x \otimes_{\mathcal{O}_{X,x}} \kappa(x)$? $\kappa(x)$ is the residue field…
phabocraph
  • 117
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Higher direct image and local cohomology.

Let $X$ be an scheme, $Z \subset X$ a closed subscheme, and $\mathcal{F}$ a coherent sheaf then, $\mathcal{R}^{i-1}_{j_{*}}(\mathcal{F}|_{X-Z})\cong\mathcal{H}_{Z}^{i}(X,\mathcal{F})$ I would like to see this isomorphism explicitly. Since I dont…
Andrea
  • 57
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Showing a morphism of affine varieties is surjective

Let $X$ and $Y$ be affine varieties such that the coordinate ring $A(Y)$ is a subring of $A(X)$. Let $$\pi:X\to Y$$ be the morphism induced by the inclusion $A(Y)\subseteq A(X)$. I need to show that $\pi$ is surjective if it satisfies the following…
user323124
  • 43
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Let $S \subset \mathbb{P}^3$ be a quartic containing a line $l$, $H$ a hyperplane section of $S$, then $|H-l|$ is a pencil of elliptic curves.

Reading the proof of Proposition VIII.15 in Beauville's "Complex algebraic surfaces", I got stuck with the following fact he is using: Let $S \subset \mathbb{P}^3$ be a quartic containing a line $l$, $H$ a hyperplane section of $S$, then $|H-l|$ is…
baltazar
  • 1,537
4
votes
1 answer

The morphism defined by a linear system associated to a smooth curve of genus 2 on a $K3$ surface has degree 2 and its branch locus is a sextic.

This is part of Proposition $VIII.13$ in Beauville's "Complex algebraic surfaces": Let $S$ be a $K3$ surface and $C \subset S$ a smooth curve of genus $g=2$. Then the morphism $\phi$ defined by the linear system $|C|$ is of degree 2, whose branch…
baltazar
  • 1,537
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Picard group of the grassmannian

I would like some help in finishing this argument, as sketched in Dolgachev's book "Lectures in Invariant Theory" (page 165). Everything here is done over $\mathbb{C}$. The claim is that $Pic(Gr_{k,n}) \cong \mathbb{Z}$. The argument goes as…
Elliot
  • 998
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A divisor on a smooth curve such that $\Omega$ minus it has no nonzero sections

Let $X$ be a smooth projective curve of genus $g$. Let $\Omega$ be the sheaf of differentials. Mumford (in Abelian Varieties, sec. 2.6, in proving the theorem of the cube) asserts that there is an effective divisor of degree $g$ on $X$ such that…
Akhil Mathew
  • 31,310
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1 answer

Exercise 4.2, chapter II in Hartshorne's Algebraic Geometry

I have trouble with the exercise mentioned: Let $S$ be a scheme, let $X$ be a reduced scheme over $S$, and let $Y$ be a separated scheme over $S$. Let $f$ and $g$ be two $S$-morphisms of $X$ to $Y$ which agree on an open dense subset of $X$. Show…
An Hoa
  • 1,543
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1 answer

How to handle group schemes by points?

I find it is very inconvenient to handle group schemes by its defination(i.e. everything is defined by morphism). And I have noticed that for group varieties, one can treat them as actual groups(i.e. talking about elements). Moreover, in the book…
Li Zhan
  • 2,673
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Where are sheaves in the functor of points perspective?

Let $X$ be a scheme over a ring $S$, and $F$ some quasicoherent sheaf on $X$. The "functor of points" says that we can think of $X$ as the functor representing all of the $R$-points, for $R$ an $S$-algebra. This recovers the classical point of view,…
Elle Najt
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