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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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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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19 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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22 views

Affine if noetherian and the reduced is affine

Prove that if a scheme $X$ is noetherian and $X_{red}$ is affine then $X$ is affine.
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28 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
19 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
38 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
40 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
29 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
37 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
29 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
24 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
47 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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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
38 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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1answer
24 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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31 views

Sheaf of 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
42 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
30 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
59 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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19 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
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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
24 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
31 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? Would running ...
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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
57 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
30 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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Why locally free implies free for graded modules? [on hold]

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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A closed subset $Z⊂\operatorname{Spec}R$ is irreducible if and only if $Z=Z(p)$ for a uniquely determined prime ideal $p$.

Definitions I am working with: $R$ is a commutative ring with $1$. A topological space $X$ is called irreducible if every decomposition $X=X_1∪X_2$ in closed subsets $X_i$ implies that either $X_1$ ...
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2answers
85 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 ...
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0answers
47 views

Kahler differentials of a Hopf Algebra

Let $A$ be a $k$-Hopf Algebra over some ring $k$, with augmentation ideal $J_A=$ ker $(\epsilon:A\rightarrow k)$ I would like to prove that the module of Khaler Differentials $\Omega_A$ of $A$ over $...
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1answer
47 views

Finiteness of intersection numbers

I'm trying to understand Shafarevich's definition of intersection numbers: By definition, "$D_1,...,D_n$ general position at $x$" means that $\bigcap_{i=1}^n\text{Supp}(D_i)$ has finitely many ...
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42 views

Why does a pointed surface minus a countable set of points contain a curve?

Let $S$ be a surface over $\mathbb{C}$ and let $s_1,\ldots, s_n$ be closed points of $S$. We consider this data as fixed. It is not hard to see that there is a curve passing through $s_1,\ldots,s_n$....
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Definition of action of Lie algebra of an algbraic group

Here is the context of my interrogation : Let $G$ be an affine algebraic group over $\mathbb{C}$ acting rationally on an affine variety $X$ over $\mathbb{C}$. This induces an action of $G$ on $\...
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41 views
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Semisimple linear algebraic group

Let $G$ be a linear reductive algebraic over an algebraically closed field of characteristic $0$. I know that there is a surjective map with finite Kernel $$G' \times T \to G$$ where $G'$ is ...
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1answer
62 views

Functor of points of the completion of a ring

Let $R$ be a ring, with ideal $I$, and let $\widehat{R}_I$ be the completion $\varprojlim R / I^n$ of $R$. Can I somehow describe the functor $\mathrm{Hom}(\widehat{R}_I,?) : Rings \to Sets$ in an ...
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28 views

Why Bosch define $O_X$-module sheaf associated to M on open basis?

The following is a quotation from Bosch「Algebraic Geometry and Commutative Algebra」(6.6 Theorem 5) Let $A$ be a ring, $X = Spec \ A$ its spectrum, and $M$ an $A$-module. Then the functor $$ \...
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1answer
26 views

If $K$ is a field then $\frac{K[x_1,x_2,…,x_n]}{(x_1-\alpha_1,…,x_i-\alpha_i)}\cong K[x_{i+1},…,x_n]$

If $K$ is a field and $\alpha_1, \alpha_2, ..., \alpha_p \in K$ then $$\frac{K[x_1,x_2,...,x_n]}{(x_1-\alpha_1,...,x_i-\alpha_i)}\cong K[x_{i+1},...,x_n] $$ My Attempt I simply tried to use Hilbert'...
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Smooth curves as iterated smooth hyperplane sections

Let $k$ be an algebraically closed field. Can every connected smooth projective one-dimensional $k$-scheme be obtained by taking iterated smooth hyperplane sections of the image of a projective ...
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0answers
20 views

On the definition of degree of closed subschemes

$\underline {Background}$:We know that,for a projective variety $X \subset\mathbb{P}^{n}=(\mathbb{K}^{n+1}-{0})/\sim$ we define , degree($X$)=$(r!)$.(leading coefficient of the hilbert polynomial of ...
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1answer
41 views

If $R$ is a PID then $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$? [on hold]

Is the following statement true? If $R$ is a PID then $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$ I know the reverse is false.
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1answer
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(Proof verification) $\pi: X\to Y$ birational with $X$ smooth implies $\pi_*O_X=O_Y$.

As the title suggests, I am looking for a confirmation or falsification of my proof of the following Let $\pi: X\to Y$ be a birational map of algebraic varieties over the complex numbers with $X$ ...
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1answer
30 views

Existence of function equal to 1 at a point in the complement of an algebraic set [on hold]

If $v\subset A^n(k)$ is an algebraic set and $(\beta_{1}, \beta_{2},...,\beta_{n}) \in A^n(k) - V$ then there exists $f\in k[x_{1},x_{2},...,x_{n}]$ such that $$\forall(\alpha_{1}, \alpha_{2},...,\...
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0answers
76 views

Proving that a prime ideal is principal

Suppose $Q_1, Q_2\in \mathbb{C}[X_0,\dots,X_n]$ are irreducible homogeneous quadratic polynomials such that $V(Q_1, Q_2)$ is an irreducible projective variety of degree two and codimension two in $\...
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2answers
63 views

Question about linear algebraic groups split vs isotropic

I am reading notes on linear algebraic groups and I'm getting confused with some definitions and I would appreciate any clarification. They define $G$ to be split if there exists a maximal torus $T$ ...
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0answers
37 views

Gluing functions from irreducible components of a reduced curve if they agree on the intersection points

The question is: What is the algebraic machinery/reasoning behind the following intuition? Given a reduced curve over some field $k$ and a regular function on each of its components such that those ...
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1answer
25 views

Tangent lines throught a point in an algebraic curve

In Fulton's Algebraic Curves, beginning of chapter 3, there is a quick discussion about multiple points and tangent lines. He gives many examples of affine curves, for instance: He then says that ...
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1answer
21 views

A problem in multispace $\Bbb P^{n_1}\times \Bbb P^{n_2}\times …\times \Bbb P^{n_r}\times \Bbb A^m$

Consider the multispace $$M:=\Bbb P^{n_1}\times \Bbb P^{n_2}\times ...\times \Bbb P^{n_r}\times \Bbb A^m.$$Let $$f\in k[x_{11},...,x_{1n_1},x_{21},...,x_{2n_2},...,x_{r1},...,x_{rn_r},y_1,...,y_m]=k[...
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0answers
50 views

Singular maps in into projective spaces

This may be a very basic question, so my apologies if that is the case. But I was interested in having some examples of meromorphic (singular) maps into complex projective space from complex surfaces ...
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0answers
37 views

Lines in projective and affine space

I'm trying to understand lines in affine and projective space in order to solve problems 2.15 and 4.13 in Algebraic Curves by William Fulton: https://www.google.com/url?sa=t&source=web&rct=j&...