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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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Show that the projective closure of V(x) and V($x-y^4-z^4$) in $P^3$ is not isomorphic

This is exercise 3.4.4 from "An Invitation to Algebraic Geometry": Show that the affine varieties V(x) and V($x-y^4-z^4$) is isomorphic in $A^3$ but their projective closures are not in $P^3$. What I did: They're isomorphic in $A^3$ since they're…
Xipan Xiao
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General nonreduced member of irreducible linear system without fixed component.

Let $X$ be an (classical) Enriques surface. Let $|D|$ be a linear system which has no fixed part. $|D|$ is irreducible and $P_a(D)>1$. If the general member of $|D|$ were not reduced, then by Bertini's theorem we would have that the general member…
Abeon
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A simple sheaf computation

I am currently taking my first course in algebraic geometry and am stuck at te following problem, which I am sure is simple. Consider $Y := \mathbb P^1 \times \{x\}$ as a closed subscheme of $\mathbb P^1 \times \mathbb P^1$. Suppose you have a…
Anonymous
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For any irreducible variety $X$ and any point $x$ in $X$, dim$\mathscr{T}(X)_x \geq$ dim$X$, with equality holds in a dense open subset of $X$

For any irreducible variety $X$ and any point $x$ in $X$, $\mathrm{dim}\mathscr{T}(X)_x \geq \mathrm{dim}X$, with equality holds in a dense open subset of $X$. Here, $\mathscr{T}(X)_x$ denotes the tangent space of $X$ at $x$. Let $\mathscr{O}_x$…
ShinyaSakai
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pullback and pushforward of line bundles

I have two questions. For the first, we consider a projective morphism between two smooth, projective varieties over $k$, $f:X\rightarrow Y$. Let $\mathcal{L}$ be a line bundle and $f^*\mathcal{L}$ the pullback. Under what assumptions is…
user109227
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$\operatorname{Spec}A$ is a finite set if and only if $A$ is a finite dimensional vector space

Let $k$ be an algebraically closed field, and $A$ be a finitely generated $k$-algebra with no nilpotents. Show that $\operatorname{Spec}A$ is a finite set if and only if $A$ is a finite dimensional vector space over $k$.
user122999
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On the Grothendieck ring of varieties

The Grothendieck group of varieties $K_0(\textrm{Var}_k)$ over a field $k$ is the Abelian group generated by isomorphism classes of quasi-projective $k$-varieties, subject to the scissor relation (under which $[Y]=[Z]+[Y\setminus Z]$ for every…
Brenin
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Identifying the stalk of the general point of an integral scheme with the field of fractions of any open affine

Let $X$ be an integral scheme and $\eta$ its general point. Then we can identify $\mathcal O_{X,\eta}$ with $FF(A)$ where $\operatorname{Spec} A$ is any open affine of $X$, because $\eta$ lives in all open affines. But how can we then extend any $f…
Rodrigo
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Why $\mathcal{O}_{\mathbb{P}^n}(1)$ is a line bundle?

Why $\mathcal{O}_{\mathbb{P}^n_{\mathbb{C}}}(1)$ is a line bundle? In the book of Hartshorne, $\mathcal{O}_{\mathbb{P}^n_{\mathbb{C}}}(1)$ is defined by $\mathcal{O}_{\mathbb{P}^n_{\mathbb{C}}}(1) = \tilde{S(1)}$, where $S=\mathbb{C}[x_0, \ldots,…
LJR
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Correspondence between the projective space associated to a vector space and the dual space of the vector space?

Let $V$ be a vector space over $\mathbb{C}$ and $V^*$ be its dual space. Let $\mathbb{P}V$ be the projective associated to $V$. It is said that homogeneous coordinates of $\mathbb{P}V$ correspond to elements of $V^*$. What is the correspondence?…
LJR
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Vanishing along a subvariety implies vanishing

Suppose that $Y \subset X \subset \mathbb P^n$ is a sequence of inclusions of closed subschemes. I'd like a result to the effect that if a polynomial of some fixed degree vanishes on $Y$, it necessarily vanishes on all of $X$. For example, if $Y$…
Mark
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ampleness of invertible sheaves

Let $f: X\rightarrow Y$ be a morphism of schemes over a field $k$. Let $\mathcal{L}$ be an invertible sheaf on $Y$. My question is If $\mathcal{L}$ is ample, is $f^*\mathcal{L}$ ample? If 1. is not true, is there the condition that…
Sang Cheol Lee
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A generalization of Cayley-Bacharach Theorem

This is exercise 19.4.B on Ravi Vakil's notes. Let $C$ be a regular plane curve of degree $e>2$, and $D_1,D_2$ be two plane curves of same degree $d$ not containing $C$. By Bezout's theorem $D_i$ and $C$ meet at $de$ points. Suppose both $D_i$ meet…
Gazerun
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Correspondence Between Cartier Divisor and Weil Divisor (Hartshorne Proposition 6.11, Chapter 2)

Hartshorne gives a correspondence between Cartier divisors of X and Weil Divisors of X, when X is integral separated locally factorial noetherian scheme. I understand given a Cartier Divisor how to define a Weil Divisor. But I don't understand the…
Babai
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Help in this easy example in algebraic geometry

It's a silly example, maybe I miss something. I will begin with a theorem in basic algebraic geometry that states: Let $f:X\to Y$ be a finite morphisms of affine varieties with $Y$ normal. Thus, for each $y\in Y$, we have $|f^{-1}(y)|\le…
user42912
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