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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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$f^{\ast}\mathcal{F}\otimes_{\mathcal{O}_X}f^{\ast}\mathcal{G}\cong f^{\ast}(\mathcal{F}\otimes_{\mathcal{O}_Y}\mathcal{G})\quad$?

Let $f$ be a morphism of schemes $f: (X,\mathcal{O}_X)\to (Y,\mathcal{O}_Y)$, and $\mathcal{F},\mathcal{G}$ be sheaves of $\mathcal{O}_Y$-modules. I am trying to prove (I do NOT claim this to be…
Li Zhan
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The normal bundle of the twisted cubic

Let $C\subset \mathbb P^3$ be the twisted cubic given by the ideal $I=(xz-y^2,yw-z^2,xw-yz)$. I want to compute the normal bundle $N_{C/\mathbb P^3}$, i.e. the dual of $\mathcal I/\mathcal I^2=(I/I^2)^\sim$. My goal is to find $h^0(N_{C/\mathbb…
Brenin
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Complements of hypersurfaces in a projective space is affine.

Suppose $H_0$ is the hypersurface defined by a homogeneous polynomial $H$ in $\mathbb{P}^n(k)$. How do we show its complement $\mathbb{P}^n(k)_H$ is affine? (It is a problem in Mumford's Redbook Ch1.5)
user93417
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Testing whether a hypersurface is singular

If one has a one variable polynomial then the discriminant can be used to test whether the polynomial has any repeated roots or equivalently where the polynomial and its derivative have a repeated root. Now If I look at a hyper-surface (Let us say…
CPM
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Curve minus finite number of points affine

I am doing another exercise from Liu. let X be a smooth geometrically connected projective curve over a field k of genus $g \geq 2$ Show that there exist at most $(2g-2)^{2g}$ points $x \in X(k)$ such that $X \setminus x$ is an affine plane…
Tedar
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Algebraic geometry "20 questions"

I'm trying to see if it's possible to do an "algebraic geometry 20-questions game" On an index card there is printed the equation for some algebraic variety $W$, in this case, let's say it's the zero-set of $x^{7}y^{3} - y^{7}z^{3} + z^{7}x^{3} =…
graveolensa
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What is the tensor product of a sheaf and a module?

The following object was studied in section III.12 of Hartshorne's Algebraic Geometry book. Let $A$ be a noetherian ring, $Y=\mathrm{Spec}A$, and $M$ be an $A$-module. Let $f:X\to Y$ be a morphism, and $\mathcal{F}$ be a quasicoherent sheaf on…
Jiangwei Xue
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A functor $(Sch)^{op}\to(Sets)$ can not represented by a scheme?

This is M. Olsson's book "Algebraic Spaces and Stacks". Exercise 1.D.(b): 1.D.(a) Let $$\mathbb{A}^n-\{0\}:(Sch)^{op}\to(Sets)$$ be a functor sending a scheme $Y$ to the set of $n$-tuples $(y_1,...,y_n)$ of sections of $\Gamma(Y,\mathscr{O}_Y)$ such…
WakeUp-X.Liu
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Bertini's theorem: reduction to pencils

I'm studying the proof of Bertini's theorem on "Principles of Algebraic Geometry" by Griffiths and Harris (page 137). The statement is as follows: The generic element of a linear system is smooth away from the base locus of the system. The authors…
Francesco Genovese
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"Higher" tangent space

Let $$ k[\varepsilon] = k[t]/t^2 $$ be the algebra of dual numbers i.e. $\varepsilon^2 = 0$, here $k$ is a field. Then for a scheme $X$ over $k$ and $x \in X(k)$ we may consider the set $X(k[\varepsilon])_x$ of $k[\varepsilon]$-valued points…
boxdot
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Showing equivalence of two definitions of the Blow-Up of a Variety

Let $X=\mathop{\mathrm{Spec}}(A)$ be an affine variety over some algebraically closed field $\Bbbk$ and $I\subseteq A$ an ideal of $A$. There are two ways to define the blow-up $\tilde X$ of $X$ along $I$, namely Set $\tilde X :=…
Jesko Hüttenhain
  • 14,792
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Singular and Sheaf Cohomology

Let $X$ be a complex manifold of dimension $n$. Thus, it's a real manifold of dimension $2n$. Now cohomology is a topological concept so it should not depend upon the structure given on a topological space. We know that $k^{th}$ Singular cohomology…
User3568
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Hartshorne proof of adjunction formula proposition II.8.20

On page 182 Hartshorne argues that $\omega_X \otimes \mathcal O_Y = \omega_Y \wedge^r (\mathcal I / \mathcal I^2)$, where Y is a nonsingular subvariety of codimension r in the nonsingular variety X over k, and $\omega$ is the canonical sheaf. Then,…
Rodrigo
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geometrically integral fibres

Let $X$, $Y = \mathrm{Spec}(A)$ be Noetherian schemes and $f: X \to Y$ be proper with geometrically integral fibres. I want to show this implies $\mathcal{O}_Y = f_*\mathcal{O}_X$. My idea was to reduce to $A$ local, use the theorem on formal…
user5262
  • 1,863
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Showing the support of a sheaf may not be closed (Liu 2.5)

This is question 2.5 of Qing Liu. I am new in algebraic geometry and really stuck on it and can't do anything to solve it. The question: Let $F$ be a sheaf on $X$. Let $\operatorname{Supp} F=\{x\in X:F_x\neq 0\}$. We want to show that in general,…
user468730
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