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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2answers
64 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
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0answers
22 views

PGL_n transformation fixing a set of n+2 points in general position must be identity

I was wondering if anybody can think of a slick and short argument to show that any $PGL_n(k)$ transformation $A$ that fixes a set of $n+2$ points $p_1, p_2,\dots, p_{n+2}$ in general position must be ...
0
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0answers
20 views

Field of definition of an Ideal

I am trying to prove the following statement from Introduction to commutative algebra and algebraic geometry by Ernst Kunz p,16 Q9 Let $I$ be an ideal of the polynomial ring $K[X_1,X_2,...,X_n]$ over ...
2
votes
1answer
26 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 ...
2
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1answer
54 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 ...
61
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0answers
1k views
+500

Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
0
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0answers
54 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 ...
2
votes
1answer
30 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 ...
4
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1answer
87 views

Fulton Algebraic Curves, Exercise 2.32(a).

Let $R$ be a DVR satisfying the conditions of Problem 2.30. Any $z \in R$ then determines a power series $\lambda_i X^i$. If $\lambda_0, \lambda_1, \dots$ are determined as in Problem 2.30(b). Show ...
4
votes
2answers
218 views

Is a morphism between smooth varieties smooth if fibers are?

Suppose that $X$ and $Y$ are smooth varieties over a field $k$ (not necessarily algebraically closed), of dimension $m$ and $n$. Suppose we are given a morphism $\pi:X\rightarrow Y$. We know that if ...
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2answers
36 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
36 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 ...
0
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2answers
67 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 ...
2
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1answer
31 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 ...
0
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1answer
15 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 ...
0
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0answers
27 views

Genus of a curve [closed]

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 ...
6
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1answer
183 views
+50

Geometric intuition for the Stein factorization theorem?

What is the intuition behind the Stein Factorization Theorem? I understand that it was originally a theorem in several complex variables, so I was wondering if there's some geometric explanation that ...
2
votes
1answer
33 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
1answer
31 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
votes
1answer
60 views

Rank 2 vector bundle

$E$ is a rank $2$ vector bundle. Why is $E\simeq E^*\otimes \det E$? Any generalization (arbitrary rank, $E$ non locally free etc.)?
2
votes
1answer
53 views

Composition of morphisms of locally ringed spaces

I have a specific question about defining the composition in (locally) ringed spaces. The definition I had formulated myself while reading Hartshorne, since he conveniently neglected to suggest any ...
3
votes
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
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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
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0answers
38 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 ...
1
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0answers
32 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 ...
2
votes
1answer
53 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 ...
2
votes
1answer
33 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
0answers
24 views

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

Where I can find a complete description of the vector bundles of $\mathbb{P}^2$?
1
vote
1answer
59 views

Direct product of algebraic groups is an algebraic group

I am attempting to verify that the product variety $G \times G'$ of algebraic groups with the direct product group structure is an algebraic group, though I'm running into trouble. In particular, I'm ...
0
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0answers
28 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
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0answers
30 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 ...
1
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0answers
46 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
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1answer
30 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 ...
1
vote
1answer
34 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 ...
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 ...
0
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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
56 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 ...
-1
votes
0answers
33 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 ...
0
votes
1answer
17 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 ...
0
votes
2answers
31 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 ...
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0answers
37 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 ...
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0answers
31 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
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1answer
35 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 ...
7
votes
1answer
78 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 ...
20
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4answers
2k views

What is a good book to study classical projective geometry for the reader familiar with algebraic geometry?

The more I study algebraic geometry, the more I realize how I should have studied projective geometry in depth before. Not that I don't understand projective space (on the contrary, I am well versed ...
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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
1answer
55 views

Quotient $G \to G/N$ induces quotient $H \to H/N$ by restriction?

Let $G$ be a linear algebraic group over an algebraically closed field $k$. Consider closed subgroups $N \subseteq H \subseteq G$ such that $N$ is a normal subgroup of $G$. Then restricting the ...
0
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0answers
31 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 ...
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 ...