Mathematics Stack Exchange
Mathematics Stack Exchange

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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Is the image of a projective variety projective?

Suppose $X$ is a projective variety and $f:X\to Y$ a morphism, is the image $f(X)$ projective?(the schematic image is well-defined in this case, or the induced reduced structure on the closed subset $f(X)$) Hartshorne says this property holds for…
user93417
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The Weil divisor corresponds $\mathcal{O}(1)$ in Proj$A[x_0,x_1,...,x_n]$

Let $A$ be an integral domain, let $\mathbb{P}^n_A=\operatorname{Proj}A[x_0,x_1,...,x_n]$, in this case, the Weil, Cartier divisor, and invertible sheaves can be used interchangeably. I have seen some heuristic argument that $\mathcal{O}(1)$…
Li Zhan
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Vector bundles on elliptic curves

Let $F$ be a stable vector bundle of degree $d$ and rank $r$, with $(r,d)$ coprime and $X$ an elliptic curve. I know that I can construct an extension $$0 \to H^0(F) \otimes O_X \to G \to F \to 0 $$ such that the boundary map of the associated long…
ArthurStuart
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Rank two vector bundle on $\mathbb P^1$, with trivial canonical bundle

I would like to show that the total space $X$ of the rank $2$ vector bundle $E=\mathscr O_{\mathbb P^1}(-1)\oplus\mathscr O_{\mathbb P^1}(-1)$ on $\mathbb P^1$ has trivial canonical bundle $\omega_X$. So we have the structure morphism…
Brenin
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What can we say about the image of a regular map?

If we have an irreducible subvariety of complex affine space $A_{\mathbb{C}}$, is the image under a regular (i.e., given by polynomials) map also irreducible? is it an irreducible subvariety of the target space?
math1234567
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Prove that Something is an Affine Variety (elementary)

To prove that a given set is an affine variety, I want to first relate it to an ideal. If there exists an ideal that is finitely generated, is that enough to prove something is an affine variety (for example, every singleton set)? Also, how would…
math1234567
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Intuition for blowing-up and the Rees algebra

Starting from an informal understanding of blowing-up as replacing a subscheme by the possible directions into it (or some more accurate formulation of this), how does one justify the definition of the blow-up of $X$ along $Y$ by the formula $$…
Will
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Pullback of a linear system in a ramified cover (through an example)

I am trying to understand the pullback of sections of a line bundle, in the situation of a ramified cover $\pi:X\rightarrow Y$ of algebraic varieties. I am trying to work out some nice concrete examples first, as the following one with a K3 surface…
Heitor Fontana
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if a property holds on closed points of an algebraic variety, does it hold over all geometric points?

Say I've got a variety X (or a scheme locally of finite type) over an algebraically closed field k. Then closed points of X correspond to k-points of X. (correct?) Let's define a geometric point of X as a morphism from an algebraically closed field…
Jacob Bell
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What is $\operatorname{Proj}$ intuitively?

For a project, I am reading a paper that assumes that the reader knows a great deal more algebraic geometry than I do (I've just begun studying the subject). I need to understand what $\operatorname{Proj}$ is, at least intuitively, so that I can go…
Daniel Robert-Nicoud
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An algebraic Möbius strip

I came up with a pair of strange 3-folds, I was wondering if there is any way to show such thing should not happen just by looking some of the properties they have. Let $V_1, V_2$ be smooth, projective, 3-folds over complex numbers. I know $V_1,…
Li Yutong
  • 4,065
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The dimension of the punctual Hilbert scheme

Let $H$ be the punctual Hilbert scheme of $3$ points in $\mathbb A^3$, over the complex numbers. Then $H$ can be described in the following equivalent ways: set-theoretically, $H$ is the set of subschemes $Z\subset \mathbb A^3$ of length $3$…
Brenin
  • 14,072
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Variety consists of complex matrices whose power is $0$

Consider the set of $n\times n$ complex matrices $X\subseteq \mathbb{C}^{n^2}$ such that $X^n=0$. What is a minimal set of at most $n^2-1$ polynomials $f_1,\ldots,f_m$ such that the set of matrices is the variety of $f_1,\ldots,f_m$? I don't really…
JJ Beck
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Some notes, literature on grassmanians

Can anybody provide a link to some notes on grassmanians? I mean something 'elementary': description of Plucker embedding, lines and hyperplanes on grassmanians and so on.
user74574
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2 answers

Function field of an affine hypersurface

I am reading Hulek' Elementary Algebraic Geometry, p.103 Let $V$ be an irreducible affine hypersurface, say $V=V(f)\subset\Bbb{A}^n$. Then the coordinate ring is by definition $k[V]=k[x_1,\ldots,x_n]/(f)$. Suppose $f$ contains the variable $x_1$.…
Heitor Fontana
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