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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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Do there exist regular functions on the projective space over a field?

Suppose $k$ is a algebraically closed field,$\mathbb{P}_k^n$ is the projective space over $k$,given the Zariski topology.A regular function on $\mathbb{P}_k^n$ is a function $f:\mathbb{P}_k^n \longrightarrow k$ which is locally a quotient of two…
user14242
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$f^*f_*(\mathcal{F})$ is surfective if $\mathcal{F}$ is generated be global sections

Suppose $\mathcal{F}$ is a sheaf of module on $X$,$f:X\to Y$,suppose $\mathcal{F}$ is generated by global sections. Is $f^*f_*(\mathcal{F})\to \mathcal{F}$ is surjective ? To check on stalks, $f^*f_*(\mathcal{F})_x \cong f_*(\mathcal{F})_{f(x)} \to…
user93417
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Prime divisor of a variety

Notation: $k$ is an algebraically closed field. By a variety I mean a separated ringed space $(X,O_X)$ that is locally isomorphic to $(Z,\mathcal O_Z)$ where $Z\subseteq\mathbb A^n_k$ is a closed Zariski subset and $\mathcal O_U$ is the structural…
Dubious
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Are restrictions of an (algebraic) vector bundle over $X \times \mathbb{A}^1$ to different copies of X isomorphic?

Practically the first property of vector bundles encountered in topology is that the pullbacks of a bundle over $Y$ by homotopic maps $X \to Y$ are isomorphic over $X$ (for reasonable spaces). Now let's consider similar situation in the algebraic…
Dmitry
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explicit display of morphisms between projective varieties

Suppose $X,Y$ are closed subvarities of $\mathbb{P}^n,\mathbb{P}^m$ Can every morphism $f:X \rightarrow Y$ be written as $(x_0,\ldots,x_n)\rightarrow (f_1(x),\ldots,f_m(x))$ where $f_i$ are homogeneous polynomials? It is the right locally on $X$,…
user93417
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Can gluing of morphisms of locally ringed spaces be expressed by an exact sequence?

Suppose $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ are locally ringed spaces. Then morphisms glue, that is, if $\{U_i\}_i$ is an open cover of $X$, then "to give a morphism $X\to Y$ is the same as giving morphisms $U_i\to Y$ that agree on…
Bruno Stonek
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How to show that $\Bbb{P}^n$ is birational to $\Bbb{A}^n$ but they are not isomorphic?

Let $\Bbb{P}^n$ be the projective $n$-space and $\Bbb{A}^n$ the affine $n$-space. It is said that $\Bbb{P}^n$ is birational to $\Bbb{A}^n$ but they are not isomorphic. In the case of $\Bbb{P}^1$ and $\Bbb{A}^1$. I think that we can define rational…
LJR
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Hilbert's Nullstellensatz and maximal ideals

I want to prove that every maximal ideal $m \subset \mathbb{C}[X_0,X_1]$ verifies: $m=(X_0 - \alpha, X_1 - \beta), (\alpha, \beta) \in \mathbb{C}^2$. I've read that $m=(X_0 - \alpha, X_1 - \beta)$, i.e., the ideal of every polynomial $f$, with…
Miguel Hernández
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Sheaf of differetials on $\mathbb{P}^n_A$

I am not clear how to display the sheaf of differetials $\Omega_{X/A}$ on $X=\mathbb{P}^n_A$ explicitly, What is its gobal section $\Omega_{X/A}(X)$ and section on the complement of the hyperplane $T_0=0$, $\Omega_{X/A}(D_+(T_0))$, and its stalk…
user93417
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existence of minimal resolutions

Let $X$ be a toric variety. A resolution of singularities $f \colon Y \rightarrow X$ is called minimal if for every resolution $g \colon Z \rightarrow X$, there is a morphism $h \colon Z \rightarrow Y$ such that $f \circ h = g$. I know and…
claudi
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Product of affine varieties vs. product of (quasi-)projective varieties

Suppose that $F_1,\ldots,F_r\in k[T_1,\ldots, T_n]$ and $G_1,\ldots,G_s\in k[S_1,\ldots,S_m]$ where $k$ is an algebraically closed field. Clearly $X=V(F_1,\ldots,F_r)\subseteq\mathbb A^n_k$ and $Y=V(G_1,\ldots,G_s)\subseteq\mathbb A^m_k$ are two…
Dubious
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The definition for the differential of a morphism

I am learning Linear Algebraic Groups without enough knowledge on Algebraic Groups. I see the definition for the differential of a morphism on page 42 of James Humphreys' Linear Algebraic Groups (GTM 21): Let $\phi: X \rightarrow Y$ be a morphism…
ShinyaSakai
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Formal scheme....

on studying the Harshorne book, I have some question for the formal scheme... Let $X$ be a noetherian scheme and let $Y$ be a closed subschme defined by a sheaf of ideals $\mathcal{I}$. Suppose that $\widehat{X}$ be the formal completion of $X$…
Sang Cheol Lee
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Is every variety (defined as separated prevariety) a locally closed subset of some projective space?

In Hartshorne Ch1, variety is defined to be a affine, quasi-affine, projective or quasi-projective variety. In Mumford's Red book, it was defined to be separated prevariety(gluing of a finite number of irreducible varieties). Is every separated…
user93417
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Determine the ideal of an affine variety

Let $X=\{(r^2,r^3,r^4) : r\in\Bbb R\}\subset \Bbb R^3$. Show that 1) $X$ is an affine variety. 2) Determine the ideal of $X$. Every $f\in\Bbb R[x,y,z]$, can we write $f$ in the form $f=p(xz-y^2)+q(z-x^2) +r$ , where $p,q\in\Bbb R[x,y,z]$? i)…
ziang chen
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