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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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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28 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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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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26 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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13 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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42 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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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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29 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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29 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
21 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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29 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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17 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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30 views

Topology of Isolated Singularities

In Singular Points of Complex Hypersurfaces, Milnor explains 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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55 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
41 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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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
34 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
24 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
46 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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21 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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28 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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46 views

$m_p=\{f\in \mathcal{O}_{V,p}| f(p)=0\}$, 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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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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36 views

$\text{Hom}_{\mathcal{O}_X}(\cdot, \mathcal{G})$ is 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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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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58 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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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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43 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 ...
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43 views

The set of polynomials which “cut out” smooth subsets of projective space is open and dense

Let $k[x_0,x_1,...,x_n]_d$ be a space of all forms (in other words, homogenous polynomials) of degree $d$ of variables $x_0, x_1,...,x_n$ over algebraically closed field $k$. Let's think of ...
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Looking for Severi varieties.

Let $K$ be an algebraically closed field of characteristic $0$, and let $\mathbb{O}$ be the Cayley algebra over $K$. Let $$ \mathfrak{J}_{3}=\{A\in\mathcal{M}_{3}(\mathbb{O}):A\text{ is Hermitian}\}, ...
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20 views

Intersection of affine subvarieties [on hold]

If the ideals $I_i$ define irreducible subvarieties of an affine space, can the scheme defined by the ideal generated by finitely many of the $I_i$ contain a embedded component?
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About double couver, linear system and binary quartic forms

I would like to understand the following statement, so I would be happy if someone could help me to understand why it holds. Suppose that $C$ is a nonsingular complete curve of genus 1 over a number ...
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1answer
34 views

cartier duality is a contravariant equivalence?

Can I get a reference for the statement that Cartier duality gives a contravariant equivalence from the category of finite etale group schemes to itself?
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Domnant morphism

Let $U=U(r,d)$ be the moduli space of stable vector bundles over some curve $X$, and $\Omega$ be the cotangent bundle on $U$. Let $W=\bigoplus _{i=1}^rH^0(X,K_X^i)$. The fiber of $\Omega$ over some ...
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1answer
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involution of affine space fixes a point

Let $\tau$ be a polynomial involution of affine space $k^n$ for $k$ algebraically closed, so $\tau(x_1,...,x_n)=(f_1(x_1,...,x_n),...,f_n(x_1,...,x_n))$ where $f_i$ are polynomials, and ...
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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 ...
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1answer
47 views

Closed points in projective space correspond to which homogenous prime ideals in $k[x_0,…,x_n]$

I'm trying to think about exercise 4.5.O in Vakil's notes on Algebraic Geometry. Before we defined the scheme $\mathbb{P}^n_k := \operatorname{Proj}(k[x_0,...,x_n])$ and showed that that for $k$ ...
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Does $\operatorname{Spec}$ preserve pushouts?

The spectrum-functor $$ \operatorname{Spec}: \mathbf{cRng}^{op}\to \mathbf{Set} $$ sends a (commutative unital) ring $R$ to the set $\operatorname{Spec}(R)=\{\mathfrak{p}\mid \mathfrak{p} \mbox{ is a ...
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66 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
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61 views

Image of a line or conic on Veronese surface.

This is part of Exercise 5.13 from Undergraduate Algebraic Geometry by Reid: Consider the Veronese surface $S$ defined by the map: $$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$ where ...
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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 ...