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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Variety of the ideal.

Hey I am trying to understand the inclusion $Z(I) \setminus Z(J) \subset Z(I:J)$through the standard definition of a variety (not the closure). I will be borrowing results from this answer. $$Z(I) = ...
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1answer
4 views

Set of roots of quadratic form $B(x,y,z,t)$ on the line $z=t=0$ is nonempty.

This is a proof from Section 7.1 of Undergraduate Algebraic Geometry by Reid. Suppose $S\subset \mathbb{P}^3$ is a nonsingular cubic surface, given by a homogeneous cubic $f=f(x,y,z,t)$. Consider ...
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1answer
23 views

Why is a discrete algebraic subset of $K^n$ finite?

Let $K$ be any field. If $A$ is the zero set of a polynomial $P\in K[X]$, then $A$ is finite. This follows from the fact that $K[X]$ is Euclidian, using commutativity of $K$. Now let $A\subset K^n$ ...
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1answer
23 views

Affine varieties and their ideals (part2)

On wikipedia, they talk about varieties $V,W$ and the $I(V)$ and $I(W)$ as well as the quotient ideal, $$I(V):I(W) = I(V - W)$$ Can someone show me a quick proof of the identity?
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Linear Algebraic Groups with Same Lie Algebra (Soft Question)

Let $G$ and $H$ be two linear algebraic groups over an algebraically closed field $F$ (char 0 ) such that their lie algebras are isomorphic. Now what can we say about the relation between these two ...
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Simply connected linear algebraic group

Following is what I understand regarding the simply connected linear algebraic groups afer reading some definition in Hochschild's 'Basic Theory of Algebraic Groups' : (I don't know about fundamental ...
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1answer
14 views

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$

Find the equation of locus of a point which is at a distance $5$ from $A$ $(4,-3)$. Could some explain how to solve this in detailed.
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17 views

$K$-affine $Hom$ functor relating to Polynomial Maps (Exercise in _Algebra: Chapter 0_ by Aluffi)

I'm currently learning some category theory and algebraic geometry, the basics at least. In Algebra: Chapter 0 by Aluffi, there is an exercise describing the relationship between morphisms of affine ...
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1answer
26 views

Torsion coherent sheaf on a curve has finite support

I would like to show that a torsion, coherent sheaf $\mathcal{F}$ on a regular integral curve $C$ is supported at a finite number of closed points. This is from Ravi Vakil's notes, namely part 13.7.G. ...
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50 views

Topology of the complex curve $x^4+y^4=1$

How do you realize that the complex curve $x^4+y^4=1$ looks topologically like three tori glued together with four points at infinity?
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Integers characterizing singularities of algebraic curves

The question in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...
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20 views

Question on singularity of variety $X$ being irrelevant of choice of polynomials defining $X$

I am getting quite confused with the following material and I would greatly appreciate if someone could provide me an explanation for this. Suppose I have $F_1, ..., F_r \in \mathbb{Q}[x_0, .., x_n]$ ...
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140 views

Are existentially defined subsets of affine algebraic sets unions of a finite number of affine algebraic sets?

Consider a set of polynomials in $\mathbb{C}[x_1,\dots,x_n]$. The zero locus of these polynomials $Z$ is a subset of $\mathbf{A}^n$ and is an affine algebraic set. Now, consider the following subset ...
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29 views

Monomorphism in the category of schemes

Let $(f, f^{\#}): X \rightarrow Y$ be a map of schemes. The stacks project gives a criterion for $f$ to be a monomorphism (see lemma 25.23.6): if (a): $f$ is a monomorphism in the category of ...
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3answers
55 views

A line avoiding an Algebraic group

Let $K$ be an algebraically closed field, and $G\subset (K,+)^3$ an algebraic subgroup (i.e. given as the zero sets of finitely many polynomial equations) of dimension 1. Is it clear that there is a ...
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34 views

How to show for a f.g. graded ring $R$, $R^{(m)}$ is generated by degree $1$ for some $m$?

Let $$R=\oplus_{i\geq 0} R_i$$ be a graded ring, which is finitely generated as a $R_0$ algebra. Let $R^{(m)}$ be $\oplus_{i\geq 0} R_{mi}$. Then how to show that for some $m \in \mathbb{N}$, ...
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1answer
28 views

Proving a projective quadric is nonsingular

Let $K$ be an algebraically closed field of characteristic $\neq 2$. Let $C$ be an irreducible quadric curve in $\mathbb{P}^2$, i.e. $C = Z(F)$ where $F$ is an irreducible degree 2 form. I think we ...
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1answer
16 views

Why does the commutator subgroup of a unipotent algebraic group have smaller dimension?

Suppose $U$ is a unipotent linear algebraic group. Is there an explanation why the commutator subgroup $[U,U]$ has strictly smaller dimension, or at least why it is a proper subgroup? This fact is ...
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1answer
50 views

Hartshorne Exercise II.2.18(d)

The Exercise: Let $\phi: A \rightarrow B$ be a ring homomorphism and let $X = \operatorname{Spec} A, Y = \operatorname{Spec} B$. Let $f: Y \rightarrow X$ be the morphism of schemes induced by $\phi$. ...
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24 views

Use Gröbner bases to count the $3$-edge colorings of planar cubic graphs…

I found a nice introduction on how to Use Gröbner bases to construct the colorings of a finite graph. Now my graphs $G=(V,E)$ are the line graphs planar cubic graphs, so they are $4$-regular. The ...
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39 views

Unramified morphism

I was reading the following page: https://ayoucis.wordpress.com/2014/04/06/unramified-morphisms/ and there are several things I do not understand and would like to clarify. First doubt The ...
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1answer
30 views

Is the Locus circle?

Locus of points such that sum of it's distances of them from four fixed points remains constant? Is the locus circle? I was not able to solve it as there were four radicals. Is it a theorem?
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31 views

Can this extension of fields be transcendental?

Let $(R, \mathfrak m)$ be a local integral domain which is contained in a field $K$. Let $0 \neq x \in K$ be such that $\mathfrak m R[x]$ is a proper ideal of $R[x]$ (one can show for any $x$ that ...
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The induced map $\phi: \mathbb{C}^n \to \mathbb{C}^m$ in the construction of toric varieties

Let $\Sigma(1)$ denote the set of one dimensional cones in a fan $\Sigma$. The corresponding vectors in the lattice are denoted $(v_1, \ldots, v_n)$ and to each $v_i$ we associate a homogeneous ...
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1answer
32 views

Principal open sets of affine schemes

This question is a special case of Open subschemes of affine schemes are affine? where it is established that in general, open subschemes of affine schemes are not affine. I was wondering if this was ...
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36 views

The projective space is not affine (II)

This question is closely related to Projective space is not affine. I want to show that the projective space is not affine and to this end I want to prove that $\Gamma(\mathbb P^n_R, \mathcal ...
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20 views

Higher direct images along the blowup

Let $S$ be a smooth projective surface and $p:X\to S\times S$ be a blowup along the diagonal with the exceptional divisor $E$. How to compute $Rp_*\mathcal{O}_X(-2E)$?
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36 views

Topology of Isolated Singularities

In Singular Points of Complex Hypersurfaces, Milnor shows that if $f: \mathbb{C}^{n+1} \rightarrow \mathbb{C}$, holomorphic, has an isolated singularity at the origin, then $f^{-1}(\epsilon) \cap ...
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0answers
57 views

On algebraic groups of dimension 1

I am searching for a possible analogue of a result in algebraic groups in a non-commutative setting, so I am looking for different proofs of the following : Let $K$ be an algebraically closed field. ...
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1answer
43 views

What is a divisor (of an algebraic curve)?

So if I have a polynomial $p(x,y)$ and define a curve $C$ based on $p$, what is a divisor? In the context I'm looking at (where I'm trying to learn about Goppa codes), in Joyner et al.'s "Applied ...
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31 views

Lie algebra of a connected simple algebraic group is simple and a simply connected algebraic group having the same Lie algebra

Let $G$ be a connected simple algebraic group over an algebraically closed field $C$. What I infer from this definition is that the defining polynomials of $G$ have coefficients in $C$ while $G$ may ...
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1answer
36 views

Is $X(k')$ a subset of $X$?

Let $X$ be an algebraic $k$-scheme in the sense of these notes (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), and let $k'$ be a field containing $k$. Let $(Y, \mathcal O_Y)$ be the $k$-scheme ...
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1answer
25 views

How to view an inclusion of $k'$-rational points

Let $X$ be an algebraic $k$-scheme in the sense of these notes (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), and let $k'$ be a field containing $k$. By $X(k')$, we mean the set of morphisms of ...
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1answer
28 views

Torsion in cotangent sheaf of the cusp

Let $X$ be the cuspidal curve $y^2 = x^3$. This is a singular curve with a unique singularity in the origin. One can easily see this using the Jacobi criterion. How does one show explicitly that the ...
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1answer
47 views

Injectivity of associate map of affine scheme homomorphism

Let $R$ be a ring and the corresponding $(\text{Spec } R, \mathcal{O}_{\text{Spec } R})$ be the affine scheme where $\mathcal{O}_{\text{Spec } R}$ is the structured sheaf of rings. By definition, the ...
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1answer
25 views

Definition of schematically dense

In these notes on algebraic groups (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), assume $(X, \mathcal O_X)$ is an algebraic $k$-scheme for some field $k$, and $S$ is a subset of $X(k)$, where ...
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Smooth points of the secant variety with a given tangent space

Let $X\subseteq\mathbb{P}^{N}$ be an algebraic variety of dimension $n$. Let $(x,y)\in X\times X-\Delta_{X}$ and $z\in\langle x,y\rangle\subseteq SX$, where $SX$ is the secant variety of $X$. I want ...
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29 views

Is this complex vector bundle trivial?

Let $\Sigma$ be any Riemann surface, and let $L \rightarrow \Sigma$ be a complex line bundle (which is classified according to its degree). Then the vector bundle $L \oplus L^{-1} \rightarrow \Sigma$ ...
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44 views

Can 2 equations defining the same curve be put together?

I have the following relation $$X -1\quad 0 \quad 2 \quad 3$$ $$Y -8 \quad 3 \quad 1 \quad 12$$ I can define the relation as $\begin{array}{rcrl} 11X-Y&=&-3&\text{ for $X$ between $-1$ ...
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1answer
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$m_p=\{f\in \mathcal{O}_{V,p}| f(p)=0\}$, ideal of $p$ in the local ring. What is $m_p/m_p^2$?

In Section 6.8 of Undergraduate Algebraic Geometry by Reid, the author proved the following Theorem: There is a natural isomorphism of vector spaces $(T_pV)^*\cong m_p/m_p^2$ where $^*$ denotes ...
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1answer
26 views

How do you find $∠XPC$ + $∠XPB$ such that $PB+PC$ is maximum where $P$ is a point on $f(x) = (x-1)(x-3)(x-5)$?

Problem: $f(x) = (x-1)(x-3)(x-5)$ intersects the x axis at $A(1,0)$, $B(3,0)$ and $C(5,0)$. A point $P(t,f(t))$ is selected on the curve such that $PB+PC$ is maximum and $t \in (3,5).$ Let $PX$ be ...
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1answer
26 views

On dimension of algebraic sets

Let $k$ be an algebraically closed field and $m\leq n$. Suppose $\pi:\mathbb{A}^n\to \mathbb{A}^m$ is map which sends $(a_1,\ldots,a_n)\to (a_1,\ldots,a_m)$. If $V$ is an affine algebraic set, then ...
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1answer
49 views

$\text{Hom}_{\mathcal{O}_X}(\cdot, \mathcal{G})$ is an exact functor

Suppose that $(X, \mathcal{O}_X)$ is a ring space. If $\mathcal{F}, \mathcal{G}$ are sheaves of $\mathcal{O}_X$-modules then the assignment $$U \mapsto \text{Hom}_{\mathcal{O}_X|U}(\mathcal{F}|U, ...
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1answer
42 views

Local properties of morphisms of schemes

In Hartshorne Proposition II.5.8, he shows, given a morphism $f \colon X \to Y$ where X and Y are schemes and $\mathcal{G}$ a quasi-coherent sheaf of $\mathcal{O}_{Y}$- modules, that ...
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1answer
54 views

Is this ring extension flat?

Let $k$ be a field of characteristic zero and let $A$ be a finitely generated $k$-algebra. Let $B=A[x_1,\ldots,x_n]$ be the polynomial ring over $A$ and let $I \subseteq B$ be an ideal such that $B/I$ ...
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1answer
61 views

Calculating (co)limits of ringed spaces in $\mathbf{Top}$

Let $\mathbf{Top}$ be the category of topological spaces, $\mathbf{RS}$ the category of ringed spaces and $\mathbf{LRS}$ the category of locally ringed spaces. There are forgetful functors $$ ...
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1answer
30 views

Two questions on surface fibrations

Let $X$ be a smooth surface and $f\colon X\to B$ a fibration, with $B$ a smooth curve. (Q1) Why is the normal bundle of any fiber $F$ trivial? It is clear to me that it has to be of degree zero ...
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1answer
46 views

Understanding the Gluing axiom of the Structure Sheaf on $Spec(R)$

Let $X = Spec(R)$ be an affine scheme for some commutative ring $R$. The structure sheaf $\mathscr{O}_{X}$ is a contravariant functor (I think) $\text{Open}(X) \leadsto \text{Ring}$ from the category ...
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36 views

Common Intersection Point of ellipsoids

Suppose we have two ellipses in $2$-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have $4$ points of intersection. Is it known ...
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1answer
50 views

functor from complex algebraic variety to constructible function

I am reading MacPherson's paper "Chern Classes for singular varieties". Proposition1 : There is a unique covariant functor from compact complex algebraic variety to abelian groups whose value on a ...