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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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Why is the disjoint union of a finite number of affine schemes an affine scheme?

We know that the disjoint union of an infinite number of affine schemes is not an affine scheme since the underlying topological space is not quasi-compact. But how do you show that the disjoint union of a finite number of affine schemes is an…
user45955
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Exact sequence in weighted projective spaces to compute Chern character

In Hartshorne's Algebraic Geometry we have Thm. II.8.13. Let $A$ be a ring, let $Y = \mathrm{Spec}(A)$, and let $X = \mathbb{P}_{A}^{n}$. Then there is an exact sequence of sheaves on $X$, $$0 \rightarrow \Omega_{X / Y} \rightarrow…
rla
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The image of a morphism between affine algebraic varieties.

Suppose $F$ is an morphism between algebraic variety $V$ and $W$. Prove that the pull back $F^\#$ between the coordinate ring $C[W]$ and $C[V]$ is surjective if and only if the morphism $F$ is an isomorphism between $V$ and some algebraic subvariety…
Li Xinghe
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Projective closure: How to determine?

In the exercise 2.9 of the book Algebraic Geometry by Hartshone, the author questions us about the projective closure of an affine variety. Let $Y$ be an affine variety in $\mathbb{A}^n$, identifying $\mathbb{A}^{n}$ with the open subset $U_0$ of…
Arsenaler
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(Problem 2.18: Algebraic curves, William Fulton) - Correspondence between prime ideals of coordinate ring and subvarieties

I am a math graduate student, and I'm working through Fulton. This is my first exposure to algebraic geometry. I'm having trouble with problem 2.18: Let $\mathcal{O}_P(V)$ be the local ring of a variety $V$ at a point $P$. Show that there is a…
Michael Porter
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How to understand whether two distinguished open sets are isomorphic

Let $R=k[x_1,...,x_n]$ be the polynomial ring over an algebraically closed field $k$ and let $f,g\in R$. Assume that $f$ and $g$ are irreducible. How can I understand whether $k[x_1,...,x_n,\frac{1}{f}]$ and $k[x_1,...,x_n,\frac{1}{g}]$ are…
Levent
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Hartshorne Exercise II. 3.19 (a)

Let $X$ be a noetherian space. We say a subset $Z$ of $X$ is constructible in $X$, if it is a finite union of locally closed subsets of $X$. There is the following theorem of Chevalley(we are not supposed to prove it in this thread). Theorem of…
Makoto Kato
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What can be said if $f^*\omega_Y = \omega_X$?

Let $f : X \to Y$ be a finite morphism of connected, reduced, pure-dimensional projective schemes of equal dimension satisfying $f^*\omega_Y \cong \omega_X$. What can be said about $f$ in this case? Is $f$ surjective? Etale? I am failing to come up…
Mellon
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Is local isomorphism totally determined by local rings?

If $X$ and $Y$ are two varieties and the germs of regular functions $\mathcal O_{x,X}$ and $\mathcal O_{y,Y}$ of two points $x \in X$ and $y \in Y$ are isomorphic as $k$-algebras, then can we find two neighborhoods $x\in U$ and $y \in V$ and an…
Summer
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About open subsets of affine schemes.

Let $A$ be a commutative ring with unity. Consider its associated affine scheme $(\operatorname{Spec}(A),\mathcal{O}_A)$. I was wondering if the restriction morphism, $$A \xrightarrow{r|^X_U} \mathcal{O}_A(U)$$ would induce an open immersion of…
Abellan
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All the lines on the Segre quadric

Find all the lines on the quadric surface in $\mathbb P^3$ defined by the equation $$xw=yz$$ (with the homogeneous coordinates $[x:y:z:w]$ of course). Now it is well known that by the Segre embedding there is an isomorphism $\mathbb P^1…
user223794
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When is a quasi-projective variety affine?

By an affine variety I mean a variety that is isomorphic to some irreducible algebraic set in $\mathbb A^n$ and by a quasi-projective variety I mean a locally closed subset of $\mathbb P^n$, with the usual Zariski topology and structure sheaf. (I am…
YZhou
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The equivalence of two definitions of closed subscheme, Vakil's Ex 8.1.K

Generally in literature, the definition of a closed embedding in the category of scheme is a morphism $\pi:X \rightarrow Y$ between two schemes such that $\pi$ induces a homeomorphism of the underlying topological space of $X$ onto a closed subset…
Wenzhe
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Smooth complete intersection counterexample

Does anyone know of a nice example of a non-singular complete intersection $X$ (say in $\mathbb{P}^n_k$, maybe even $k=\overline{k}$, char($k$)=0) such that it cannot be written as $E\cap H$ where $E$ is a non-singular complete intersection and $H$…
Matt
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What is the difference between algebraic sets and algebraic varieties?

I am wondering what is the difference between algebraic sets and algebraic varieties in complex projective space. It seems that both are zero sets of polynomials, so what is the difference?
user326227
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