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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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23 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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10 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
23 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 $y\...
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0answers
25 views

The scheme is affine if it is noetherian and its reduced scheme is affine

Let $X$ be a scheme with its structural sheaf $O_{X}$.We define a subsheaf of ideals $N_{X}$ of $O_{X}$ as the sheafification of the presheaf which associates to each open subset $U \subset X $ the ...
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1answer
17 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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24 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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36 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
27 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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41 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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14 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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20 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
26 views

Workaround for Windows Magma Memory Limit

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
44 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
48 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
26 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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27 views

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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25 views

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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1answer
51 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
30 views

Differentials of Hopf Algebras

Let $A$ be a $k$-Hopf Algebra over some ring $k$, with augmentation ideal $I$. I would like to prove that the module of Khaler Differentials $\Omega_A$ of $A$ over $k$ is isomorphic to the tensor ...
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1answer
46 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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0answers
41 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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0answers
28 views

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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0answers
20 views

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 ...
2
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1answer
60 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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25 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
25 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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0answers
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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
19 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
38 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
29 views

(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
74 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
61 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
32 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
20 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
47 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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35 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&...
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0answers
35 views

What does vanishing of higher direct images of the structure sheaf tell us?

Let $f:X\to Y$ be a morphism of schemes over $k$. I am wondering about, what geometric consequences $R^qf_*O_X=0$ for $q\geq k$ does have. I saw vanishing of higher direct images used in some proofs ...
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1answer
122 views

A map is injective if it is nonzero at the generic point?

Proposition IV 2.1 in Hartshorne's states that if $f: X\to Y$ is a finite separable morphism of curves. Then $f^{*}\Omega_Y\to \Omega_X$ is injective. And he proves this by saying that it will be ...
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2answers
42 views

How is this classical group $\textit{compact}$? [duplicate]

Let $O(n)$ be the group of orthogonal $n \times n$ matrices. Apparently this is a "compact classical group" but I have trouble seeing that it is compact. The topology is the topology is inherits from $...
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1answer
34 views

Isomorphism of canonical rings and arithmetic genus.

Suppose you have two integral Gorenstein projective curves $X$ and $Y$ over a field, and suppose further that we have an isomorphism $$\bigoplus_{n=0}^\infty H^0(X,\omega_X^{\otimes n}) \cong \...
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0answers
33 views

Direct image of structure sheaf of a singular curve

Suppose $f:X\rightarrow C$ is a degree two morphism of projective curves over complex numbers. Assume $C$ is smooth and $X$ has only nodal singularities at $d$ points and smooth elsewhere. So the ...
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0answers
35 views

Prove that if k is a field, then any affine open subset of the affine n-space A_n^k is principal.

I think the question is clear and famous too but i didn' find an answer on this site, so : k : is a field not necessarily ( an algebraic closed field ). An open subset of the scheme is an open in ...
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0answers
22 views

Expressing the Hilbert polynomial of a complete intersection as the difference of two Hilbert polynomials

Suppose $ Z_{i} = \text{Proj}[k_{0},\dots,x_{n}]/(F_{i}) $ is a hypersurface of degree $ d_{i} $ in projective space over a field $ k. $ Consider $ \Gamma = \text{Proj}[k_{0},\dots,x_{n}]/(F_{1},\...
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1answer
60 views

On solution to the equation $x_{1}^{2}x_{2}^{2}x_{3}^{2}x_{4}^{2}x_{5}^{2}\left(\sum_{i=1}^{6}a_{i}x_{i}\right)^{2}=1$

For any $a_{1}, a_{2}, \dots, a_{6} \in \mathbb{R}$ with $$\sum_{i=1}^{6}a_{i}^{2}=1$$ is it true that there always exist $x_{1}, x_{2}, \dots, x_{6} \in \mathbb{R}$ with $\displaystyle\sum_{i=1}^{...
1
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1answer
55 views

Proving Pascal's theorem from a corollary to Max Noether's Fundamental theorem

A result in my book Fulton's Algebraic Curves states that: If $F$ and $G$ meet in $\deg(F\deg(G)$ distinct points, and $H$ passes through these points then there exists a curve $B$ such that $B•F = H•...
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0answers
40 views

Structure sheaf and residue field

Hartshorne defines the structure sheaf of $X=$ Spec$A$ as, for $ U \subset X$, $$ O_X(U)=\{s: U \rightarrow \dot\cup A_{\mathfrak{p}} \}$$ with 2 additional requirements, i.e. $s({\mathfrak{p}}) \in ...