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. ...

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2
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1answer
16 views

Affine variety over a field which is not algebraically closed

I am now trying to prove the following statement. If the field $K$ is not algebraically closed, then any $K$-variety $V\subset\mathbb{A}$ can be written as the zero set of a single polynomial in ...
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45 views

Proving this fact about algebraic sets.

I want to prove the following equivalence: Let $V$ an algebraic set, $K$ a field and $\overline K$ its algebraic closure. Then we say that $V/K$ ($V$ is defined over $K$) if $I_{V}$ (the ideal ...
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0answers
16 views

Plücker relations for $Gr_2(\mathbb{C}^5)$

I know that for the complex Grassmannian $Gr_2(\mathbb{C}^4)$, with the index sets $I= \{1,2\}, J = \{3,4\}$, the Plücker relation is given by $$ P_{12}P_{34} - P_{32}P_{14} - P_{13}P_{24} =0 \text{,} ...
2
votes
1answer
45 views

Fulton , Algebraic Curves, Exercise 2.15

I'm doing some self study on Fulton's Algebraic Curves, and I've done a decent amount. But I'm stuck on a past question that's been bugging me. Question : Let $K$ be a field, and $P,P'$ be points in ...
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2answers
27 views

How to prove that any regular map $\phi : \Bbb P^1 \to \Bbb A^n(\Bbb C)$ maps $\Bbb P^1$ to point.

How to prove that any regular map $\phi : \Bbb P^1 \to \Bbb A^n(\Bbb C)$ maps $\Bbb P^1$ to point. Now $\phi=(F_1/G_1,...,F_n/G_n)$ where $F_i/G_i$ is a regular function. Now how do I conclude?
1
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1answer
25 views

Projection of the twisted cubic

Question: Let $X$ be the Twisted Cubic in $\mathbb{P}^3$, and $\pi_p:X\rightarrow \mathbb{P}^2$, the projection of the Twisted Cubic from $p$. Find the equations of the projection of the twisted cubic ...
7
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38 views

Are weakly étale ring homomorphisms of finite presentation étale?

Following [Stacks, 092A], say a ring homomorphism $A \to B$ is weakly étale if both $A \to B$ and $B \otimes_A B \to B$ are flat. Question. Are weakly étale ring ...
0
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1answer
12 views

An example of geometrically integral projective surface that is not smooth

Let $X \subseteq \mathbb{P}^n$ be a geometrically integral projective surface of degree $d$ defined over $\mathbb{Q}$. Does it then imply that $X$ is smooth? I was wondering if someone could provide ...
2
votes
1answer
17 views

Multiple fibres of an elliptic fibration on Enriques surfaces

Let $X$ be an Enriques surface and $f\colon X\to \Bbb{P}^1$ an elliptic fibration. I will denote by $F$ the general fibre and by $r_1F_1,\cdots,r_kF_k$ the multiple fibres of the fibration. I wanted ...
2
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1answer
27 views

can a field be embedded into product of hyperbolas as a closed subset?

Let's say we have a algebraically closed field $k$. Let $H=\{(x_1,x_2)\in k^2 : x_1x_2=1\}$ be an affine variety. My question is, can $k^1$ be embedded into $H \times H$ as a closed subset? My idea ...
2
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1answer
25 views

Quasicoherent sheaf on the functor of points is the same as on the scheme itself

I've seen the definition of a quasicoherent sheaf $\mathcal{F}$ on an arbitrary functor $$ X : CRing \to Sets $$ as a specification of an $R$-module $\mathcal{F}(x)$ for each $R$-point $x \in X(R)$ ...
0
votes
2answers
49 views

Affine varieties and their ideals

I was reading on Wikipedia about quotient ideals. It mentions that if $W$ and $V$ are affine varieties (assume $V$ is) and $I(V)$ and $I(W)$ are the ideals for $V$ and $W$, then $$I(V):I(W) = ...
0
votes
1answer
21 views

Self-intersection number in Projective Space

The question is based on the example given in Intersection Theory under the heading Self-intersection. The example is as follows: Consider a line $L$ in the projective plane $\mathbb{CP}^{2}$: it has ...
3
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0answers
17 views

How the normal bundle of a divisor changes under a (fnite) quotient map

Let X be smooth and projective, D is an Cartier divisor of X. $\mathcal{N}_{D/X}$ is the normal bundle of D in X. Let $q: X\longrightarrow Y$ be a finite quotient map of degree d. And q is totally ...
1
vote
1answer
37 views

There exists a zero dimensional ideal I such that $\dim (R/I) - |V(I)| \geq \alpha > 0$

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). I know that ...
2
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0answers
36 views

Why is a projective variety 'the best kind'?

In Hartshorne's AG, he discusses the classification of curves by birational equivalence class says 'based on the idea that a nonsingular projective variety is the best kind..'. What exactly makes a ...
2
votes
1answer
27 views

Why is $|V(I)| \leq d_1\cdots d_n$?

If $I \subset K[x_1,\dots,x_n]$ is a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). Then if $G$ is a ...
0
votes
2answers
57 views

how to prove that an algebraic variety is normal but not smooth?

Let $X=\{(x_1,x_2,x_3) \in \mathbb C^3 : x_1^2=x_2^2+x_3^2\}$, an algebraic variety. How do i prove that $X$ is normal, but not smooth? I guess the non-smoothness appears at the point $(0,0,0)$, but I ...
1
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1answer
19 views

Infinitely many non-isomorphic degree 8, dimension zero schemes in the plane

In Geometry of Schemes by Eisenbud and Harris, it is claimed in Exercise II-19 that: There are infinitely many isomorphism types of degree 7 subschemes supported at the origin in 3-space, and ...
2
votes
1answer
51 views

Is the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ birational?

This is Exercise 5.3 (a) in Undergraduate Algebraic Geometry by Reid. Does the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ define a rational map? Determine ...
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0answers
24 views

vector bundles of $\mathbb{P}^2$ [on hold]

Where I can find a complete description of the vector bundles of $\mathbb{P}^2$?
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0answers
32 views

Equivariant generalization of $\mathcal{O}(1)$ on $\mathbb{CP}^1$

The connection of the line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^1$ is given by \begin{equation} A=\frac{i}{2}\frac{\overline{z} \, dz-z\,d\overline{z}}{1+|z|^2} \end{equation} This follows since ...
0
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0answers
24 views

Genus of a curve [on hold]

Let $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{D}$ are given. I have tried to find the genus of the following curve, and I have found that it has genus zero in $\mathbb{C}^2$, but if I restrict to ...
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0answers
29 views

Nullstellensatz to prove Noether Normalization

In many commutative algebra texts, Noether Normalization Lemma is proved and then Hilbert's Nullstellensatz is obtained as a corollary. Nullstellensatz and Normalization Lemma seem to be non-trivial ...
0
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0answers
24 views

Why the canonical bundle of a complex manifold is a line bundle?

I think I do not understand something in the definition of the line bundles. Line bundles have fibers of rank 1. That is isomorphic to $\mathbb{C}$. But I do not know how to connect this vector space ...
0
votes
0answers
28 views

Is the dual of a flat module flat

Let $k$ be an algebraically closed field, $T$ be a integral, regular, projective $k$-scheme and $X$ another projective, integral $k$-scheme. Let $\mathcal{F}$ be a coherent (pure) sheaf on $X \times_k ...
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0answers
33 views

Distance between point and ellipse - explanation of a paper

EDIT: I notice that the link is hidden, but this post is made with reference to THIS PAPER I'm trying to solve quite an old problem (once again) - to find the distance between a point (in 3d space) ...
1
vote
1answer
21 views

Which projective varieties are étale over affine space?

In Liu's answer to this MO question there is a characterization of smooth affine varieties which are étale over affine space. I was wondering if one can give a similar characterization for projective ...
1
vote
1answer
27 views

$I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}$

How to prove $I(X_1 \cap X_2)=\sqrt{I(X_1)+I(X_2)}$? Clearly $\sqrt{I(X_1)+I(X_2)} \subseteq I(X_1 \cap X_2)$ But for $f \in I(X_1 \cap X_2)$ $f(x)=0 \forall x\in X_1 \cap X_2$. how to show $f \in ...
2
votes
1answer
53 views

Why is even codimension necessary to apply excision for the Euler characteristic?

In this answer on MathOverflow, it is claimed that $$\chi(X/Z)=\chi(X)-\chi(Z)$$ holds for complex subvarieties $Z$ only because $Z$ has even codimension. It is implied that for $Z$ with odd ...
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votes
0answers
47 views

Spectrum and maximal spectrum of a ring

How do the $\mathrm{Spec}(\mathbb{C}\left [ X \right ])$ and $\text{m-Spec}(\mathbb{C}\left [ X \right ])$ look like? I understand the definitions of $\mathrm{Spec}(R)$ and $\text{m-Spec}(R)$ for a ...
2
votes
1answer
30 views

Explicit procedure for integrating densities (twisted $n$-forms)?

I'm aware of several slightly different (yet equivalent) definitions for the density bundle of a non-orientable manifold. Unfortunately, I've been unable to make sense of integration in any of them. ...
0
votes
1answer
16 views

Prove that graph of $f$ , $G(f)$ is an affine variety.

Let $f \in \tau(V)$, $V$ a variety in $\Bbb A^n$. Define $G(f)=\{(a_1,\ldots,a_{n+1})\in \Bbb A^{n+1} \mid (a_1,\ldots,a_n)\in V$ and $a_{n+1}=f(a_1,\ldots,a_n)\} $ Prove that $G(f)$ is an affine ...
1
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0answers
33 views

Coherent sheaves on $\mathbb{P}^1$

Let $F$ be a coherent sheaf on $\mathbb{P}^1$. How to show that there exists a unique exact sequence of the form $$0\to\mathcal{O}_{\mathbb{P}^1}(-1)^{\oplus a}\to\mathcal{O}_{\mathbb{P}^1}^{\oplus ...
0
votes
2answers
28 views

If $\phi ^{-1}(X)$ is irreducible, and $X$ is contained in the image of $\phi$, show that $X$ is irreducible.

If $\phi: V \to W$ is a polynomial map, and $X$ is an algebraic subset of $W$, show that $\phi ^{-1}(X)$ is an algebraic subset of $V$. If $\phi ^{-1}(X)$ is irreducible, and $X$ is contained in the ...
0
votes
1answer
28 views

How to prove that $\dim_k k[V]< \infty$ implies $V$ is a point.

Let $k$ be an algebraically closed field. Let $V \subset \Bbb A^n$ be a nonempty variety. How to prove that $\dim_k k[V]< \infty$ implies $V$ is a point. I am not getting the answer even I do not ...
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0answers
29 views

Buchberger's Algorithm Example

I've been reading Ideals, Varieties and Algorithms and came across an example of Buchberger's algorithm being computed and I am not able to understand how they came to have the final result. The ...
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0answers
27 views

Group action of linear algebraic group $G$ on itself induces a representaion of $G$ on $Lie(G)$

Let us be given a linear algebraic group $G$ over a field $K$ of characterstic zero. This group $G$ is defined as the common zeroes of a finite set of polynomials $\{f_1, \ldots ,f_r\}$ $\in K ...
0
votes
1answer
33 views

Action of an algebraic group induce a representation of its Lie algebra

Let $G$ be a linear algebraic group over a field $K$ of characterstic zero acting on a vector space $V$. Then does this action induce a representation : $$\Gamma : Lie(G) \to gl(V)$$ If yes, how ? ...
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0answers
21 views

The coordinate ring of $\varepsilon: xy-1=0$ [duplicate]

I want to show that the coordinate ring $\mathbb{R}[x,y]/\mathbb{R}[\varepsilon]$ of $\varepsilon: xy-1=0$ is not isomorphic with the polynomial ring of one variable $\mathbb{R}[x]$. To me this is ...
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0answers
29 views

Relation between

Can anybody please give a relationship among these objects. Varieties, schemes, moduli spaces, stacks, algebraic spaces, groupoids. I mean here, we can define an abstract Variety as a scheme with ...
7
votes
1answer
76 views

How does Hartshorne's definition of group schemes encode the law for the neutral element?

Hartshorne's Algebraic Geometry says A scheme $X$ with a morphism to another scheme $S$ is a group scheme over $S$ if there is a section $e\colon\;S\to X$ (the identity) and a morphism ...
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0answers
34 views

What does $F^n$ mean in the case of fields?

I am reading notes for Hilbert's Nullstellensatz and came across this expression. $$\exists F, \mathcal{V}_F(I):\{\xi \in F^n:f(\xi)=0, \forall f\in I\}\neq \emptyset$$ Here, $F=\mathcal{P}/m$ is a ...
1
vote
2answers
35 views

Prove that all automorphisms of the line $\Bbb A^1$ are of the form $f(x) = ax + b$ with $a\neq 0$.

An isomorphism $f : X → X$ of a closed set $X$ to itself is called an automorphism. Prove that all automorphisms of the line $\Bbb A^1$ are of the form $f(x) = ax + b$ with $a\neq 0$. I think I can ...
0
votes
2answers
33 views

Show that $K(U) \cong K(X)$

Let $X$ be an irreducible algebraic set. Let $U$ be an Zariski open subset of $X$. Show that $K(U) \cong K(X)$, where $K(X) = $Set of rational function on $X$. Obviously if $U \subsetneq X$. This is ...
0
votes
0answers
18 views

Example where the closure of a point does not contain a closed point in a Nonnoetherian Scheme

Let $x\in X$ be a point of a Noetherian scheme. Then the closure of $x$ contains a closed point. Is there an example in non-Noetherian schemes where this property does not hold?
1
vote
1answer
41 views

Is it true that for any set $X,Y \subseteq \Bbb A^n(k)$, $I(X) = I(Y )$ iff $X = Y$ .

Is it true that for any set $X,Y \subseteq \Bbb A^n(k)$, $I(X) = I(Y )$ iff $X = Y$ . Because I know $I(X) = I(Y )$ implies $\overline X=\overline Y$ But I am thinking that this is not true because if ...
1
vote
0answers
24 views

Does this base change yield another dominant morphism?

Here's something that seems to be true, or at least I hope it to be true, but I'm unable to prove it: Let $S$ be a $k$-rational surface and $B$ a curve, both projective, smooth and geometrically ...
2
votes
0answers
21 views

Pic of a variety of type G/P

Let $G$ be an simple algebraic group an let $P$ be a parabolic subgroup of $G$. Let $X$ be the projective, homogeneous variety $G/P$. Is it true that the following holds: Pic($X$) has rank $1$ iff ...
0
votes
0answers
15 views

Univariate residue in multivariate contour integration?

Consider a multivariate contour integral on the complex polydisc: $$\oint \frac{dz_1dz_2...dz_n}{f_1(z_1,z_2,...,z_n)f_2(z_1,z_2,...,z_n)...f_n(z_1,z_2,...,z_n)}F(z_1,z_2,...,z_n)$$ The $f_i$ in ...