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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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Local ring on generic fiber

Let $\pi: X\to C$ be a fibration in curves where $C$ is a non-singular curve and $X$ a regular, integral surface and the generic fiber $X_\eta$ is a non-singular curve over $k(C)$ (these hypotheses might be stronger than necessary, but I just threw…
Matt
  • 7,358
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Is a proper morphism between projective schemes projective morphism？

That is if $f :P^n_ {X} \rightarrow P^m_{Y}$ is a proper morphism, then it is a projective morphism. Projective morphism means that it factors through projective scheme of last term and the first component is a closed immersion.
Jian
  • 2,470
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algebraic variety of dimension 0

How to prove that if algebraic variety has zero dimension it must be finite set of points. In other words: how to prove that it can't be infinite set ??? Definitions algebraic variety V - if V is the common zero set of a polynomial system over Field…
duncan
  • 395
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Exercise I-$43$(a) in Eisenbud-Harris

This is exercise I-$43$(a) in Eisenbud/Harris, Geometry of Schemes. Let $k[t] \rightarrow k[x,u]/(xu)$ be a $k$-algebra homomorphism given by $t \mapsto x+u$. For $\alpha \in k$, the fiber ring corresponding to the prime ideal $(t-\alpha)$ of…
Manos
  • 25,833
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Prime ideal implies irreducible affine variety

Let $X$ be some affine variety. I am trying to understand why if $I(X)$ is prime, then $X$ is irreducible. In a proof here, the author states: Let $I(X)$ be a prime ideal, and suppose that $X=X_1\cup X_2$. Then $I(X)=I(X_1)\cap I(X_2)$. As $I(X)$…
Santana Afton
  • 6,758
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Relation between $\operatorname{Proj} \, k[x_0,\cdots,x_n]$ and $\mathbb{P}^n$

In Hartshorne's "Algebraic Geometry" p. 77, Example 2.5.1, it is mentioned that if "$k$ is an algebraically closed field, then the subspace of closed points of $\operatorname{Proj} \, k[x_0,\cdots,x_n]$ is naturally homeomorphic to the projective…
Manos
  • 25,833
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Quotient maps in algebraic geometry

In topology, a quotient map is a surjective map $\pi:X\to Y$ such that $V\subseteq Y$ is open in $Y$ if and only if $\pi^{-1}(V)$ is open in X. This definition has the following nice property: If $\rho:X\to Z$ is a continuous map and $f:Y\to Z$ is…
Marco Flores
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Is it possible to compute if an algebraic variety is a differential manifold?

Is it possible for a computer to decide if a given real algebraic (or semi-algebraic) variety is a differential manifold ? Let $f_1,…,f_p, g_1,…,g_q$ polynomials in $n$ variables with coefficients in $\mathbb{R}$. Let $V$ be the variety defined by…
francis-jamet
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Exercise 4.9, Chapter I, in Hartshorne

Let $X$ be a projective variety of dimension $r$ in $\mathbf{P}^n$ with $n\geq r+2$. Show that for suitable choice of $P\notin X$, and a linear $\mathbf{P}^{n-1}\subseteq \mathbf{P}^n$, the projection from $P$ to $\mathbf{P}^{n-1}$ induces a…
m-agag2016
  • 934
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A coherent-sheaf is locally free when restricted on an open dense subset, if it is globally locally free.

$X$ is a algebraic variety. $\mathcal{F}$ is a coherent sheaf. $U\subset X$ is an open dense subset. $\mathcal{F}|_U$ is locally free on $U$. Can we conclude that $\mathcal{F}$ is locally free on $X$ ?
Shuhang
  • 664
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Which class of morphisms preserves smoothness of points?

Let $f\colon X\to Y$ be a morphism of complex algebraic varieties. I am interested in those $f$ that have the following property: For any smooth point $x\in X$, the image $y=f(x)$ is a smooth point of $Y$. Certainly, open immersions have this…
Jesko Hüttenhain
  • 14,792
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Is it true that $(\mathbb{C}[x,y]/(xy))_{(x,y-k)}=\mathbb{C}[t]_{(t)}$?

I want to show that $(\mathbb{C}[x,y]/(xy))_{(x,y-k)}=\mathbb{C}[t]_{(t)}$. But I think that the leftside has 2 variables, but rightside has only 1. Is it possible that the two are isomorphic?
Gobi
  • 7,458
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The differential of the Gauss map of the Theta Divisor of a quartic plane curve.

Let's suppose we have a non-hyperelliptic Riemann surface $X$ of genus $3$ (without lost of generality, embedded in $\mathbb{P}^2$ as a smooth plane curve) and we consider its Theta Divisor. Is well known that the Gauss Map of the Theta Divisor can…
Lucke
  • 81
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Showing product of normal varieties is normal

let $X, Y$ be two affine normal varieties, over an algebraically closed field $k$, I want to show that $X\times Y$ is still a normal affine variety. There is an answer for "Does the fiber product of two normal varieties remain normal?", but I…
AG learner
  • 4,523
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Cartier divisors and non-reduced points

Let X be a non-projective non-reduced scheme and let D be an effective Cartier divisor on X. Why is D disjoint from $Ass(\mathcal{O}_X)$? In other words, why can't reduced points lie in the support of any effectice Cartier divisor? The question…
Luc
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