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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principal ideal of $\mathbb{C}[Z,\bar{Z}]$

Let $I$ a Ideal of $\mathbb{C}[Z,\bar{Z}]$. How to prove that $I$ is principal in $\mathbb{C}[Z,\bar{Z}]$ ? It exists some simple criterion to say that an ideal will be principal or not ?
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63 views

Studying the intersection $(X)\cap (X^{2}-Y+1)\subseteq\mathbb{R}[X,Y]$.

I am trying to find the intersection of ideals $$ (X)\cap (X^{2}-Y+1)\subseteq\mathbb{R}[X,Y]. $$ This is what I have tried: $$ f\in(X^{2}-Y+1)\Rightarrow f=g\cdot (X^{2}-Y+1)\text{ for certain }g\...
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1answer
49 views

Universal property, localization of rings and modules, and initial element in a category

Sorry for the confusing title. I just started learning category theory and am very confused about the concept "universal property". I am not even sure whether my "proof" is a proof or is just a ...
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2answers
44 views

Reference request: When is a conic birationally equivalent to the projective line?

I am looking for a reference which contains the proof of the following theorem: "A conic $C$ defined over the field $\mathbb{F}$ is birationally equivalent to $\mathbb{P}^{1}(\mathbb{F})$ if and only ...
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3answers
81 views

Finitely generated projective modules over polynomial rings with integral coefficients

There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free. I have never carefully ...
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18 views

Using valuative criterion of separatedness. (Hartshorne)

Hartshorne writes that for a scheme $X$ to be separated, it should not contain any subscheme which looks like a curve with a doubled point. He then writes that another way of saying the above is: ...
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1answer
40 views

Why is an exceptional sheaf on $P^n$ locally free?

Let $F$ be a coherent sheaf on $\mathbb{P}^n$ with the property that $Ext^i(F,F)$ is zero for $i > 0$, and for $i = 0$ is a one dimensional vector space over the base field. Such a sheaf is said to ...
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27 views

Inclusion Mapping?

Is the hooked arrow map notation not supposed to mean an inclusion mapping? His definition is clearly showing inputs from $\mathbb{R}^2$ who's images are elements in $\mathbb{R}^{3}$ with fixed $z$ ...
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1answer
44 views

Silverman, arithmetic of EC, I1.9 no nonconstant morphisms $P^m \to P^n$ for m>n

This topic goes about problem 9 of the first chapter of Silverman, arithmetic of EC: If $m>n$, prove that there are no nonconstant morphisms $P^m \to P^n$. A solution can be found for example at ...
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59 views

Normal bundle to an exceptional sphere in a blowup along a smooth subvariety

Let $M$ be a smooth algebraic variety of dimension $m$ over $\mathbb{C}$, $S \subset M$ a smooth embedded subvariety of codimension two. Let $M_S$ denote the blowup along $S$ and $E$ denote the ...
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19 views

How to measure the sparsity of dots on a line?

I am not sure whether there exists any method to measure the sparsity of dots on a line. This is what I think that sparsity (after linear mapping) is supposed to be: $0 < SPARSITY([s, t\ , ..., \...
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1answer
32 views

Effective divisors exactly those with global sections

Let $X$ be a finite-type scheme over a field $k$. To an effective divisor $D$, there is a global section of the invertible sheaf $\mathcal{O}_X(D)$ (corresponding to the canonical morphism $\mathcal{O}...
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25 views

Why is this an equivalent condition for stability of curves mapped to projective space?

In Fulton-Pandharipande's Notes on Stable Maps and Quantum Cohomology he claims on page 11 that if $X = \mathbb{P}^r$, the stability of a flat family of curves $(\pi:C\to S, \{p_i\}, \mu)$ where $$ \...
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1answer
71 views

Categorical Quotient and group actions

I am trying to practice calculating categorical quotients and I ran into this example. I am unable to get the answer and was wondering if someone can help? Let $G = Z/3Z =$ $\{1, \omega, \omega^2\}$, ...
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13 views

Realization of prequantized Hilbert schemes

Could we define the product of an integral scheme over an algebraic subvariety of positive characteristic if the non-reduced points are not split-solvable over the field? Perhaps a geometric ...
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95 views

Calculating Categorical Quotients [duplicate]

I am trying to practice calculating categorical quotients and I ran into this example. I am unable to get the answer and was wondering if someone can help? Let $G = Z/3Z =$ $\{1, \omega, \omega^2\}$, ...
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0answers
35 views

Basis for differentials of a smooth plane curve

Given a smooth plane curve $C$ cut out by a homogeneous polynomial $f(x,y,z)=0$, how to calculate a basis for the space of global differential forms? There is the adjunction formula which shows that ...
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2answers
56 views

How can I compute a presentation of the tangent bundle for a smooth manifold defined by a family of polynomials?

Consider a smooth manifold $M$ given by a system of polynomials $$ \begin{align*} f_1 = 0 \\ \cdots \\ f_k = 0 \end{align*} $$ in $n$ variables. This has the algebraic description as the $\mathbb{R}$-...
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1answer
17 views

Normalization of schemes which are not reduced

One usually defines normalization for reduced schemes. Is it possible to do it also for non-reduced ones? We know that to any scheme we can associate a reduced one. Is then sufficient to work on this ...
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1answer
48 views

Example of non-noetherian ring whose spectrum is noetherian and infinite

A topological space is noetherian if it satisfies the descending chain condition for its closed subsets. Let be $R$ a commutative ring and let $\mathrm{Spec}(R)$ its spectrum with Zariski topology. I ...
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31 views

Proof Check: Prove Relation Between Invariants Is the Only Relation

Consider the finite matrix group $C_{4} \subset$ GL$(2,\mathbb{C})$ generated by $$A = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \in \text{GL}(2,\mathbb{C}).$$ (a)Prove that $C_{4}$ is ...
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22 views

What properties are preserved by direct limits? [on hold]

We know that direct limit of a directed family of flat $R$-modules is also flat ($R$ is a commutative ring with $1$ and all modules are unital). I am looking for other properties of modules which ...
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29 views

Is quotient of open invariant subset open?

I am reading GIT book by Mumford. He needs special cases of the following conjecture several times. Conjecture Let $G$ be a reductive algebraic group acting on affine scheme $X=Spec A$. Let $Y = ...
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3answers
514 views

Reflection of a curve around a slant line

a fifth-degree function: y = 80*x^5-225*x^4+350*x^3-300*x^2+150*x-20 (the green curve in the image) needs to be reflected/mirrored around the line ...
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1answer
40 views

$k[x,y,z]/(y-x^2,z-x^3)\cong k[x]$, where $k$ is a field

This is generalizing from a previous question, which asks to prove that $k[x,y]/(y-x^2)\cong k[x]$. The way I proved that was by using the homomorphism $\phi:k[x,y]/(y-x^2)\to k[x]$, $\phi(\overline{f(...
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1answer
48 views

How to prove that the Lefschetz number is invariante under homotopy?

How to prove that the Lefschetz number is invariante under homotopy? We define the Lefschetz number as the number of $f : M \to M$ as the number of intersection of the map $g(x) = (x,f(x))$ with the ...
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1answer
68 views

On the definition of a reductive group.

Wikipedia defines a reductive group $G$ as an algebraic group with trivial unipotent radical. The radical is the connected component of identity in the maximal normal solvable subgroup of $G$. The ...
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19 views

Complex conics as a Riemann surface

Consider the complex curve defined by $\{(x,y) \in \hat{\mathbb{C}}^2 | ax^2 + bxy + cy^2 + dx + ey + f = 0 \}$ for some complex numbers $a,b,c,d,e,f$ (here $\hat{\mathbb{C}}$ is the Riemann sphere). ...
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1answer
31 views

Is the product of algebraic groups the same as the fibre product?

Assuming we have two algebraic groups $G_1$ and $G_2$ over $k$. Then the direct product $G_1 \times G_2$ with the direct product group structure is an algebraic group. Is this the same as the fibre ...
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1answer
79 views

Reference request: Galois descent

What is a classic (perhaps even original) reference for Galois descent? I know that it can be seen as a special case of faithfully flat descent (for which FGA and SGA I is the usual reference) and ...
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41 views

How are varieties related polynomials?

My teacher says that varieties and ideals are related to each other while I tend to mix polynomials and varieties in my terminology. Could some explain how varieties are related to polynomials? And ...
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16 views

Duality between cut ideals and cycle ideals?

There exist a general duality between vertex-cuts and cycles and also Duality Principle on Digraphs. I am trying to find a duality prienciple expressed in terms of ideals so Does there exist a ...
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1answer
58 views

Moduli Space of Hyperelliptic Curves as Fibration?

Basically, I had a thought about a way to think of the moduli space of hyperelliptic curves. I'm sure it's wrong most likely, but I was hoping someone could maybe point out the flaw in my reasoning. ...
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24 views

Example of an monomial ideal that is weakly reverse lexicographic but not reverse lexicographic

We are looking at a paper titled "Generic Ideals and Moreno-Socias Conjecture" by Edith Aguirre, et al. In the paper they state that an ideal which is reverse lexicographic is also weakly ...
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1answer
56 views

Example $3.3.1$ in Hartshorne

Let $k$ be an algebraically closed field, and let $$X = \operatorname{Spec} k[x,y,t]/(ty-x^2)$$ $$Y = \operatorname{Spec} k[t]$$ Hartshorne comments that both schemes $X$ and $Y$ are of finite type ...
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50 views

If there is a inclusion of restrictions, then is there an inclusion of sheaves?

Let $X$ be the projective space over $\mathbb{C}$. Let $H$ be a smooth hyperplane in $X$. Let $F$ and $G$ be torsion-free sheaves on $X$ of rank 1 and 2 respectively such that we have an inclusion $F|...
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1answer
68 views

Splitting a short exact sequence of complexes of vector spaces

It's well-known that any complex of vector spaces is isomorphic to a direct sum of two types of indecomposable complexes (a one-dimensional space concentrated in one degree, or two one dimensional ...
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59 views

Definition of cotangent and conormal bundle

I have read the following definition of cotangent bundle: Let $X$ be a $n$-dimensional smooth algebraic variety. For any $p\in X$ there exist a neighbourhood $U_{p}\subseteq X$ and functions (...
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48 views

Geometric and arithmetic Frobenius

I read in Serre's "Lectures on $N_X(p)$" that when $X$ is a scheme defined over $\mathbb{F}_q$ (a finite field), the geometric Frobenius $F: X \mapsto X$ is defined by fixing every element of the ...
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42 views

Algebraic Geometry Project ideas related to Computer Science

I am a Computer Science Undergrad student with an interest towards Algebraic Geometry.I have just recently started and am currently reading Miles Reids' Undergraduate Algebraic Geometry(I have read ...
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47 views

Kernel of the evaluation map of a sheaf has global sections?

Suppose that $X$ is a smooth projective variety over an algebraically closed field, for example $\mathbb{C}$. Let $F$ be a coherent sheaf on $X$. Consider a subspace $V$ of $\Gamma(X,F)$, the space of ...
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1answer
100 views

Prove that $\Bbb{R}[\cos(\theta),\sin(\theta)]\cong\Bbb{R}[x,y]/(1-x^2-y^2)$ [duplicate]

More precisely, given the ring homomorphism $\phi:\Bbb{R}[x,y]\to\Bbb{R}^\Bbb{R}$, with $\phi(f(x,y)):\Bbb{R}\to\Bbb{R},\,\,\phi(f(x,y))(\theta)=f(\cos(\theta),\sin(\theta))$, where $\Bbb{R}[x,y]$ is ...
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35 views

Extending a morphism to a proper scheme using the valuative crierion for properness

Here's an example I'm trying to work out. Let $f : U := \mathbb{A}^1 -\{0\} \rightarrow X$ be a morphism of schemes, where X is a proper scheme over a field $k$. I am trying to extend this morphism to ...
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1answer
42 views

Global section defines a map from structure sheaf

Let $X$ be a smooth projective scheme over an algebraically closed field. Let $F$ be a coherent torsion-free sheaf on $X$. A global section $f$ of $F$ defines a morphism $O_X\rightarrow F$ given by: ...
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47 views

Underlying topological space of $X_y = X \times_Y Spec \hspace{0.5mm} k(y)$.

Let $f: X \to Y$ be a morphism of schemes, and let $y \in Y$ be a point, $k(y)$ the residue field of $y$, $Spec \hspace{0.5mm} k(y) \to Y$ the natural morphism. Let $X_y = X \times_Y Spec \hspace{0....
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31 views

Ext group of bundles on moduli space of curves

Let $\mathcal{M}_{g}$ be the moduli space of curves of genus $g$. Let's suppose $g \geq 2$. Let $T$ be the tangent bundle of $\mathcal{M}_{g}$. Is the Ext group $\text{Ext}^1(\bigwedge^2T, T)$ trivial?...
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7 views

Planar ternary ring point operations

I have the following topic in my exam questions' list: Prove that point operations in a planar ternary ring satisfy field axioms. I know Proposition 1 from this paper but this only says something ...
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1answer
22 views

Endomorphism ring of an abelian variety and its reduction mod $\mathfrak{p}$

Let $A$ be an abelian variety defined over a number field $K$. Let $\mathfrak{p}$ be a prime of $K$ for which $A$ has good reduction and let $k=\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}$. Let $\...