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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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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1answer
26 views

Local sections of $\mathcal{O}(1)$

Let $\mathcal{O}(-1)$ be the tautological bundle over $\mathbb{P}^n$ and $\mathcal{O}(1)$ its dual bundle, also known as the hyperplane bundle. I know that there is a bijection between $\Gamma(\mathbb{...
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0answers
13 views

Finding valuations/uniformizers for the branches of the blow up of a singular curve

I understand that for a nonsingular curve $C(x,y)$, the uniformizer at a point $(a,b)$ is either $x-a$ or $y-b$, since the partial derivatives with respect to $x$ and $y$ are not both 0. However, if ...
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37 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 an irreducible affine scheme $X=Spec ...
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13 views

Is it enough to check corank of jacobian matrix at closed points

This is actually exercise 12.2.H of Vakil's notes. In the notes, a k-scheme is defined to be k-smooth of dimension d if there exists a affine open cover(every is of form $A=k[x_1,...,x_n]/(f_1,...,f_r)...
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1answer
35 views

Is affine GIT quotient necessarily an open map?

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite ...
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1answer
25 views

How is a dominant rational map well-defined?

I am slightly confused by the definition of a dominant rational map in Hartshorne, specifically because of a comment he makes about the equivalence relation. In Chapter 1.4, he defines a rational map ...
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18 views

Points at infinity of a conic section and its eccentricity, foci, and directrix?

Background on projective geometry and conic sections; you might want to skip to the actual question A conic section is analytically described as the zero-locus of points $(x,y)$ in the affine plane ...
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24 views

Reasoning behind definition of $\operatorname{Proj} S$.

Let $S$ be a graded rings. We denote $S_{+}$ to be the ideal $\oplus_{d >0} S_d$. We define the set $\operatorname{Proj} S$ to be the set of homogenous prime ideals $\frak{p}$, which do not ...
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26 views

When is Cartier dual of a finite group etale?

I am trying to solve the following exercise from Waterhouse: Introduction to affine group schemes (Chapter 6, Ex. 12 on page 53) without any success. Let $char(k)=p >0$ and let $G$ be an abelian ...
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1answer
94 views

Relation between stalks of twisted sheaf and structure sheaf

Let $A$ be a ring, $B = A[T_0,\dots, T_d]$, and $X = \textrm{Proj } B$. Then at every point $x \in X$, $$\mathcal{O}_X (n)_x \cong \mathcal{O}_{X,x}$$ Let $x$ correspond to a homogeneous prime ...
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32 views

pullback relation of normal bundles

We have the following setup (I don't know if my question holds in a more general setup): Let $f \colon X \to Y$ be a surjective finite morphism from a pure dimensional reduced projective variety $X$ ...
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27 views

Quasi-compact, locally of finite presentation, finite type morphisms of finite type schemes over $\mathbb{C}$

Let's say that we have two schemes $X,Y$ of finite type over $\mathbb{C}$. Question: Is is true that any morphism $f:X\rightarrow Y$ (compatible with the structure morphisms) is: 1) quasi-compact? ...
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18 views

Do there exist torsion sheaf over moduli spaces?

Usually people bother with studying moduli spaces of (coherent) torsion free sheaves that live on a topological space $X$. These spaces, actually stacks, are badly behaved topological spaces. Still, ...
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+50

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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32 views

Function as a combination of 1-forms on a Riemann surface

My question is quite simple, I hope it's not also stupid.. Consider $R$ a Riemann surface and $\omega_1$, $\omega_2$ two $(1,0)$-forms (i.e. holomorphic forms) and $\varphi_1$, $\varphi_2$ two $(0,1)$-...
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38 views

Hartshorne's algebraic geometry ; geometric understanding and intuition for intersection multiplicity

I am reading section $7$ of the book. He defines intersection multiplicity as Let $Y$ be a projective variety of dimension $r$. Let $H$ be a hypersurface not containing $Y$. Then by (7.2) $Y\cap ...
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31 views

Understanding the Definition of minimal prime ideal of a graded module

I am reading algebraic geometry from Robin Hartshorne. He has used a term "$p$ is a minimal prime of a graded $S$ module $M$". What does it mean? I know the definition of minimal prime over an ideal.
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41 views

Motivation for equivalence of Tautological Line Bundle and ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$

I currently reading a book on Complex Algebraic Geometry and both the Tautological Line Bundle $L \to \mathbb{P}_k^n$ and the blow up ${\operatorname{Bl} _0}\mathbb{A}_k^{n + 1}$ are defined set ...
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1answer
21 views

Generic translates of a divisor intersect curves

Let $G$ be a connected algebraic group and suppose $G$ acts transitively on a proper variety $X$ (say over the complex numbers). If you pick a curve $C$ and an effective divisor $D$, I have seen a ...
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1answer
50 views

Finding singular points of $x^2=x^4 +y^4$

Locate the singular point of $x^2= x^4 + y^4$, assuming that $\operatorname{char} k \neq 2$. I am using the following definition: Let $Y \subset A_k^n$ be an affine variety, and let $f_1, \dots,...
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1answer
58 views

Hartshorne IV.6.4 - no curve of degree 9 and genus 11 in P^3

I'm working on this exercise in Hartshorne: there are no curves of degree 9 and genus 11 in $\mathbb{P}^3$. The hint says to show that it would have to lie on a quadric surface. This is the part I'm ...
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1answer
60 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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1answer
39 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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0answers
50 views

Introductory Book on Faltings' Proof of the Mordell Conjecture

I'm currently reading Diamond and Shurman's book a First Course in Modular Forms and I've found it to be a wonderful introduction to the modularity theorem. Is there a similar introductory book for ...
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1answer
48 views

The ideal of the image of homogeneous polynomials

Let $k$ be an algebraically closed field, and $f_0,\dots,f_m \in k[x_0,\dots,x_n]$ be homogeneous polynomials of the same degree. Denote by $I\subset k[x_0,\dots,x_m]$ the kernel of the homomorphism ...
0
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1answer
118 views

Klein's Theorem

Let $Q$ be the projective quadric hypersurface in $\mathbb{P}^n$ defined by $x_0^2+\cdots+ x_r^2$. In Hartshorne book Ex.II.6.5, Klein's theorem is that if $r\geq 4$ and if $Y$ is an irreducible ...
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1answer
39 views

Proof that Epicycloids are Algebraic Curves?

Epicycloids are most commonly described by the parametric equations, $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$ $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right)...
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1answer
47 views

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

Let $I$ be an 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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1answer
33 views

Where are these rational functions coming from?

In the proof of the theorem below (Springer, Linear Algebraic Groups), $T$ is a maximal torus of $G$, with dimension $1$, $B$ is a Borel subgroup of $G$ containing $T$, and $U$ is the set of unipotent ...
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1answer
65 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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19 views

Describe Singular Locus of Hyperelliptic Curves?

Previously, I asked a question here: Moduli Space of Hyperelliptic Curves as Fibration? about fibering the moduli space of hyperelliptic curves $\rm{Conf}_{2n}(\mathbb{P}^{1}) \big/ \rm{Aut}(\mathbb{P}...
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3answers
98 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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3answers
64 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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2answers
50 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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0answers
29 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
49 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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31 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
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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1answer
21 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
40 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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1answer
76 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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0answers
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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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41 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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61 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
19 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
34 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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25 views

What properties are preserved by direct limits? [closed]

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 ...
2
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3answers
515 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 ...