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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.

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A question about rational functions and their divisors

Let $f: X' \rightarrow X$ be the blowing up of a nonsingular variety $X$ along a nonsingular subvariety $Z$ of codimension $r \geq 2$, and let $Z' = f^{-1}(Z)$. Denote $\mathrm{Cl}(X')$ the divisor class group of $X'$. Why is the map $\mathbb{Z}…
rla
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sections of an invertible sheaf, and their support

Suppose I have a section $s$ of an invertible sheaf $L$, vanishing along a divisor $D$. Then there is an isomorphism $(L, s) \simeq (O(D), 1)$. In the next paragraph I'll pick $D=K_X$, but that is just how I stumbled upon my question/confusion, it…
ykm
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tensor product of presheaves of modules

Let $\mathscr{O}$ be a presheaf of rings on $X$ and $\mathscr{F}$, $\mathscr{G}$ be presheaves of $\mathscr{O}$-modules on $X$. Let $\mathscr{O}^{\#}$,$\mathscr{F}^{\#}$ and $\mathscr{G}^{\#}$ be respectively the sheafification. Then is the…
nick
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Image/type of the canonical divisor under the isomorphism $\mathrm{Pic}(\mathbb{P^{1}} \times \mathbb{P^{1}}) \cong \mathbb{Z} \oplus \mathbb{Z}$

It is well known that we have an isomorphism $\mathrm{Pic}(\mathbb{P^{1}} \times \mathbb{P^{1}}) \cong \mathbb{Z} \oplus \mathbb{Z}$. Does anyone know how to determine the type of the canonical divisor $\omega_{\mathbb{P^{1}} \times…
rla
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Geometrically integral

Here is a stupid question about the notion of geometric integrality. Say I have a smooth, projective variety $X$ over a some field $k$, equipped with a morphism $f: X \to C$ to a smooth, projective curve $C$, such that the generic fibre is…
Evariste
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How to Calculate the 2 balance points "Swinging Sticks" physics toy/desk sculpture??

Please, this is something I have thought of a long time, but don't have unlimited supplies to keep doing it wrong. I want to make one of these for my nephew, he's interested in science! There is a Physics Toy, called "Swinging Sticks", used in the…
Mike Esposito
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Subtle aspect of closed subscheme

Let me define a closed subscheme of a scheme $X$ as: An equivalence class of data in the form $$(Z,Y,i,i^\sharp)$$ where $Z$ is a closed subset of $X$, $(i,i^\sharp): Y \to X$ is a morphism of schemes with $i$ giving a homeomorphism onto $Z$, and…
user51197
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Why finite morphisms between schemes are defined in a different sense of "local" with morphisms of finite type?

A finite morphism $f:X\rightarrow Y$ firstly requires an affine cover $V_i\subset Y$, such that $f^{-1}(V_i)$ are affine open sets. However a morphism of (locally) finite type $f:X\rightarrow Y$ involves a cover $V_i\subset Y$ such that…
Honglu
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Regarding a paper relating surfaces and integrable mappings

I'm reading the paper "A classification of two-dimensional integrable mappings and rational elliptic surfaces". I have two questions: Let $X$ be a generalized Halphen surface (e.g. an elliptic surface), $D = -K_X$, $\omega$ a 2-form on $X$ with…
M. E.
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Why does restriction of Weil divisors "clearly" preserve principal-ness?

(EDIT June 4, 2014: I've reposted this to MathOverflow due to lack of response here.) Let $Q$ be the subvariety of $\mathbb{P}^3$ given by $\{x_0x_1=x_2x_3\}$. Given a prime divisor $Y$ of $\mathbb{P}^3$, Hartshorne (II, Example 6.6.2) defines a…
Avi Steiner
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Quotient of a variety and orbits

Suppose a group G acts on a variety X and a quotient exists, that is, we have a variety Y and a regular map $\pi : X \rightarrow Y$ so that any regular map $\varphi :X \rightarrow Z$ to another variety Z factors through $\pi$ if and only if $\varphi…
Michi89
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base change of line bundles

Under what kind of assumptions on $X$ and $k$ (here $X$ is a scheme over $k$ and $\bar{k}$ the algebraic closure) is the map $\mathcal{L}\mapsto \mathcal{L}\otimes_k\bar{k}$ from $\mathrm{Pic}(X)$ to $\mathrm{Pic}(X\otimes_k\bar{k})$ injective?
user109227
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What is a substitute for $\textrm{Sym}^n(X)$ when $X$ is not quasi-projective?

If $X$ is a quasi-projective variety, then for each integer $n\geq 0$ one can define the symmetric product to be the scheme quotient $$\textrm{Sym}^n(X)=X^n/S_n,$$ where $S_n$ is the symmetric group and $X^n$ is $n$-fold power of copies of $X$ over…
Brenin
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Linear group action on a projective variety

I already posed this question, but my formulation was quite erroneous and unclear so I decided to repost it (which is hopefully not against the rules). On page 116 in Harris' book "Algebraic Geometry - A First Course" an action of a group $G$ on a…
Paul
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Regular maps and isomorphisms. A problem of the punctured affine plane

I am just learning the basics of algebraic geometry, mainly by reading Hulek's book "Elementary algebraic geometry". I found there a problem concerning proving that the punctured affine plane $A:=\mathbb{A}^2_\mathbb{C}\setminus\{(0,0)\}$ is not…
Damian Sobota
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