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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.

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9 views

Divisor class group 0 for affine and Z for projective nonsingular varieties given UFD

The following exercise is from Shafarevich. Let $X$ be a nonsingular affine (resp. projective) variety over an algebraically closed field. Prove that $CL(X) = 0~(\text{resp. } \mathbb{Z})$ iff the ...
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14 views

$\bigcup_{j\in J} Z(I_j) \subset Z(\bigcap_{j\in J})$

Let $\{I_j\}_{j \in J}$ be a family of ideals of $k[x_1,...,x_n]$. We have to $$ \bigcup_{j\in J} Z(I_j) \subset Z(\bigcap_{j\in J}) . $$ If $J$ is finite, so is equality. I look for an example ...
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1answer
44 views

Open Affine subsets of $\mathbb{A}^n_k$ are principal

I have heard that affine open subsets of $\mathbb{A}^n_k$ are all principal (i.e of the form $D_f$). I know this is not true if one removes the assumption the open subset is affine. I have tried to ...
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0answers
13 views

Divisors $D$ such that $(\mathbb{A}^{2}_{\mathbb{C}},\frac{3}{4}D)$ has strictly log canonical singularities

Let $P:= 0\in\mathbb{A}^{2}_{\mathbb{C}}$ and $D$ be a divisor in $\mathbb{A}^{2}_{\mathbb{C}}$. Let us assume that the pair $(\mathbb{A}^{2}_{\mathbb{C}},\frac{3}{4}D)$ has strictly log canonical ...
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0answers
42 views

$\sqrt{I} \subset I(Z(I))$

Let $I$ be a ideal of $k[x_1,...,x_2]$. Show that $\sqrt{I} \subset I(Z(I))$. Observing that $J \subset I(Z(J))$, for any ideal $J$ of $k[x_1,...,x_2]$ and that $Z(\sqrt{I})=Z(I)$, we have completed ...
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1answer
37 views

Torsion Free Module over Dedeking Ring

Let $\phi: R \to A$ be a finite morphism of Dedekind rings (so $A$ is a finitely generated $R$-module) and $M$ a finitely generated $A$-module. Obviously, if we restrict the action of $A$ on $M$ to $R$...
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1answer
25 views

Ideals of subsets in $k[x_1,..,x_n]$

Suppose that $k$ is algebraically closed field and $X_1, X_2 \subset \mathbb{A}^n_k$. Show that $$ I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}. $$ I thought the first thing to do was to use the ...
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31 views

Hausdorff Property for Preschemes in Mumford's Red Book

Let $f,g: K \to X$ two morphisms between preschemes. In order to "compare" these two morphisms in David Mumford's "Red Book of Varieties and Schemes" there is suggested (see page 118) for a $x \in K$ ...
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24 views

Torsion Free Quasi Coherent Module

Let $C$ be regular curve. Consider a finite and locally free morphism $f: C \to \mathbb{P}^1$. (the latter mean that $f_* \mathcal{O}_{C}$ is a free $\mathcal{O}_{P^1}$ module) Let $\mathcal{F}$ be ...
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24 views

Given the multiplication morphism on the scheme functor, how to get it as scheme morphism?

I am studying group schemes. I would say I understand the definition of an $S$-group scheme as to give an $S$-scheme together with morphisms for multiplication law, identity, and inverse and also the ...
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0answers
17 views

Support of Torsion Sheaf

Let $\mathbb{P}^1$ the projective line (considered as scheme) and $\mathcal{F}$ a quasi corent sheaf of finite type on on it. Denote by $\mathcal{F}_T \subset \mathcal{F}$ it's torsion subsheaf. My ...
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1answer
54 views

Hartshorne, Exercise III 4.2 (a): A morphism $\mathcal{O}^r \to f_* \mathscr{M}$ that is an iso over the generic point.

I'm having some trouble with Exercise III 4.2 a) in Hartshorne's Algebraic geometry. It is Let $f: X \to Y$ be a finite surjective morphism of integral noetherian schemes. Show that there is a ...
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0answers
18 views

Degree of a morphism is equal to the degree of a divisor?

A morphism $f:X\to P^n$ can be determined by a linear system $|D|$ for a given divisor. It seems that degree of morphism $degf$ is equal to degree of divisor $degD$. I guess the divisor may be ...
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0answers
26 views

Is the projection $p: X\times_Z Y\to Y$ also an immersion?

Let $f:X\to Z$ be an immersion of schemes and $g: Y\to Z$ is any morphism of schemes, is the projection $p: X\times_Z Y\to Y$ also an immersion?
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1answer
31 views

Kernel of pullback map of line bundles

Let $C$ be a complex projective curve with atmost nodes as singularities and let $v:C'\rightarrow C$ be its normalization. Can a non-trivial line bundle $L$ on $C$ pullback to a trivial line bundle ...
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25 views

Is $\operatorname{Id}_Y: Y\to Y$ the scheme theoretic image of $f$?

Suppose $Z$ is a Noetherian scheme and $f:Z\to Y$ is an open immersion of schemes, $f(Z)$ is dense in $Y$, is $\operatorname{Id}_Y: Y\to Y$ the scheme theoretic image of $f$?
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1answer
41 views

The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology.

The book "Invitation to Algebraic Geometry" says the following: The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology. Why is this this the case? This is thing that is asked ...
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0answers
16 views

is strict transform square of blowup cartesian

Let $X$ be a scheme and $Z$ be a closed subscheme of $X$ with sheaf of ideals $I$. Let $ \pi : Bl_{Z}X \rightarrow X$ be the blowup of $X$ at $Z$. Further let $i: Y \rightarrow X$ be another closed ...
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0answers
18 views

Extension of a rational map in codimension one - relative version

Suppose $X$ and $Y$ are smooth projective $T$-varieties, where $T$ is a smooth affine curve. Let $\phi:X\dashrightarrow Y/T$ be a rational map over $T$. My question is: is there a closed subset $Z\...
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1answer
39 views

Is $Y$ the scheme theoretic image of $f$?

Let $f:X\to Y$ be an open immersion of schemes, and $X$ is Noetherian, is $Y$ the scheme theoretic image of $f$?
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0answers
33 views

Does $f=g$ as the maps on the underlying topological spaces?

Let $f,g:X\to Y$ be two morphisms of schemes, if $U$ is a dense open set of $X$, and $f|_U=g|_U$ as morphisms of schemes, does $f=g$ as the maps on the underlying topological spaces?
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1answer
29 views

Why are morphisms of finite type defined in the usual way?

We say that a morphism of schemes $f : Y \to X$ is finite if $X$ may be covered by affine open sets $\text{Spec}(B)$ such that each $f^{-1}(\text{Spec}(B))$ is affine, say of the form $\text{Spec}(A)$,...
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0answers
67 views

Do regular functions on an algebraic variety separate points?

Define a variety as a reduced separated scheme of finite type over an algebraically closed field. Given two closed points $P$ and $Q$ on a variety, does there exists an open set containing them and a ...
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0answers
26 views

Affine if noetherian and the reduced is affine [duplicate]

Prove that if a scheme $X$ is noetherian and $X_{red}$ is affine then $X$ is affine.
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0answers
44 views

Ulrich, Torsten Proposition 8.4.2 (Closed Subscheme)

Let $S$ be a scheme and let $v : \mathcal E \longrightarrow \mathcal F$ be a morphism of quasi-coherent $\mathcal O_{S}$-modules. Let $\mathcal F$ be finite locally free. Then the locus $v=0$ is ...
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1answer
25 views

A small difficulty with the fact that a vector bundles of rank $ r $ are isomorphic to locally free sheaves of rank $ r. $

I am following the 2003 notes of Andreas Gathmann for this. A vector bundle of rank r on a scheme $ X $ over a field $ k $ is a $ k-$scheme $ F $ and a $ k-$morphism $ \pi: F \rightarrow X, $ ...
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0answers
49 views

Orlov's result on algebraic dimension

I want to know about the result of Orlov i.e., $D^b(X)≅D^b(X′)\Longrightarrow a(X)=a(X′) $. Which paper does this result appear in?
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1answer
46 views

Quotient of Algebraic Variety

Let $\mathbb{A}^1$ be the affine line (considered as algebraic variety) endowed with action by $\mathbb{G}_m$ via $t \cdot a = ta$. If we form a set theoretical quotient $\mathbb{A}^1/\mathbb{G}_m$ ...
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1answer
31 views

Prove polynomial map between two affine curves is bijection except origin

I wanna to solve this problem dealing with a polynomial map between two affine curves. First curve is $A \subset \mathbb{C}^2$ defined by equation $s(1+t^2)-1=0$ and the second curve is $B \subset \...
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1answer
41 views

Are algebraic groups defined over $\mathbb{R}$ Lie groups?

In the notes I am reading, which is about algebraic groups, in the section about over $\mathbb{R}$ all the sudden they started using the word Lie groups. I understand Lie groups and algebraic groups ...
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0answers
30 views

Why is $Hom(T(-1),O(1)) \cong \Lambda^2(\mathbb{C}^{n+1})^*$ on $\mathbb{CP}^n$?

I am currently trying to read a paper and the author claims the following. On $\mathbb{CP}^5$ we have $$Hom(T(-1),O(1)) \cong \Lambda^2(\mathbb{C}^{6})^*.$$ The proof is claimed to be a consequence of ...
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1answer
26 views

Definition of Morphism between varieties

I have found the following definition of morphism. Let $X,Y$ be two varieties. A mapping $\phi:X\rightarrow Y$ is said to be morphism if $\phi$ is continuous. For each open set $U$ of $Y$ and $f\in ...
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1answer
49 views

Are there $k$-rational points that are not closed point?

I know for a scheme $X$ locally of finite type over a field $k$, $k$-rational points are closed ponits. If we remove the assumption that $X$ is locally of finite type over $k$, are there some $k$-...
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0answers
19 views

Is the map $\bigsqcup_{w } \frac{BwP}{P} = \frac{GL_n(\mathbb{C})}{P} \to GL_n(\mathbb{C})$ via $bwP \mapsto bw$ a morphism of algebraic varieties?

Let $G = GL_n(\mathbb{C})$, $P \subseteq G$ a parabolic subgroup, and consider the decomposition $\frac{G}{P} = \bigsqcup_{w \in W^P} \frac{BwP}{P}$ where $W \cong S_n$ is the Weyl group of $G$, $W_P$ ...
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0answers
39 views

Galois cover and solvability

This is a very general question: let $f:X\to \mathbb P^1$ be a branched cover of compact Riemann surfaces, where $X\subset \mathbb P^n\times \mathbb P^1$ and $f$ is the natural projection. For any ...
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2answers
35 views

Fano Variety and Blow up

Let $Y$ be a Fano threefolds and $\pi : X \rightarrow Y$ be the blowup along a nonsingular subvariety $Z$ of $Y$ with $\mbox{codim(Z)} > 1$. Is $X$ a Fano threefold?
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1answer
85 views

Sheaf of a Closed Subset

I’ve been given the following definition: Let $(X,\mathcal{O}_X)$ be a ringed space which is locally isomorphic to an affine algebraic variety, and $Y\subseteq X$ be closed. Then for an open $V\...
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0answers
45 views

An order-6 configuration

Here's an example of an order $96_6$ configuration found by L.W. Berman. Every point has six lines, every line has six points. The unique 6,6 cage graph is bipartite and is a Levi graph for the ...
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1answer
34 views

Genus of a rational normal curve.

Let $X$ be the d-uple embedding of $P^1$ to $P^d$, for any $d\geq1$. We call $X$ the rational normal curve of degree $d$. Hartshorne's says that if $X$ is any curve of degree $d$ in $P^n$, with $d\...
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0answers
67 views

Purely technical question on the definition of a sheaf : by Yoneda + restriction or by an equalizer

I have problems dealing with all the categorical language, even for very basic things like elementary limits calculus. I always understand the intuition but I seem to be unable to right down the ...
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0answers
20 views

Divisor class group of vector bundle over an integral Noetherian scheme

Ler $X$ be an integral Noetherian scheme. Then one can show that taking the inverse image induces an isomorphism of Weil divisor class groups $\operatorname{Cl}(X)\to \operatorname{Cl}(\mathbb A^n_X)$....
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1answer
51 views

Low-degree polynomial $T\in\mathbb F[x,y]$ with $T(P(z),Q(z))=0$

Given polynomials $P,Q\in\mathbb F[z]$ over a finite field $\mathbb F$, one can find a non-zero polynomial $T\in\mathbb F[x,y]$ such that $T(P(z),Q(z))=0$ for any $z\in\mathbb F$. Is there a way to ...
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0answers
26 views

Concerning Hartshorne's proof of the Vanishing Theorem of Grothendieck (Hartshorne III 2.7)

At certain point of the proof of the theorem in Hartshorne's book, he mentioned the following in which I don't understand: In Step 3, he said we can reduce our consideration to a sheaf $\mathscr{F}$ ...
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0answers
14 views

For what fields does this morphism from an elliptic curve to the projective line ramify at infinity?

Let $k$ be a field. Consider the curve $X := V_+(X_1^2 X_2 + X_1 X_2^2- X_0^3-X_0^2X_2) \subseteq \mathbb{P}^2_k = \text{Proj}(k[X_0,X_1,X_2])$. Consider the morphism given on functions fields by $k(...
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0answers
33 views

Workaround for Windows Magma Memory Limit [duplicate]

I found this question which said that Magma for Windows has a strict memory limit of 1.3 GB as a consequence of something weird with long ints. Is there any known workaround for this on a Windows ...
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1answer
47 views

Irreducible and connected components of $\mathbb{C}[x,y]/(xy)$

There is a question which states that $\operatorname{Spec}(\mathbb{C}[x,y]/(xy))$ consists of three prime ideals: $0, (x), (y)$ I want to find irreducible and connected components of $\operatorname{...
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1answer
59 views

Notions of Fundamental Groups for semisimple algebraic groups

Let $G$ be a connected, semisimple linear algebraic group over a field $k$. In Springer's Linear Algebraic group, the Fundamental Group of $G$ is defined by a certain quotient of groups $P/Q$, which ...
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0answers
31 views

Surjection from a scheme of finite type

Let $S$ be a scheme, $X$ a scheme locally of finite type over $S$, $Y$ a scheme over $S$. Let $f:X\rightarrow Y$ a surjective morphism of $S$-schemes. Under what conditions is $Y$ locally of finite ...
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0answers
33 views

Why locally free implies free for graded modules? [closed]

Suppose $S_\bullet=\oplus_{i=0}^\infty S_i$ is a graded noetherian ring with $S_0$ is a field. Let $M$ be an $S_\bullet$-graded module. If $M$ is locally free, then why $M$ is free?
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2answers
99 views

Kernel of an algebra map and module of Kahler Differentials

Let $A$ be a $k$-algebra, $f:A\rightarrow k$ an algebra map with kernel $I$. I'd like to prove that $\Omega_A\otimes_f k $ is canonically isomorphic to $I/I^2$. This is from W.C.Waterhouse Intro to ...