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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Under what type of transformations are characteristic classes and characteristic numbers of a manfold invariant?

When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they ...
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26 views

Shouldn't $t^n : \mathbb{A}^1 \rightarrow \mathbb{A}^1$ ramifies at $0$?

Yo, this is probably the stupidest question ever that I've asked here. Let $$\varphi: \mathbb{A}^1 \rightarrow \mathbb{A}^1$$ be the map of schemes (over a field $k$) such that $\varphi (x) = x^n$. ...
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19 views

Is there a “strong” Chow lemma where “dense” means “scheme theoretically dense”?

Recall Chow's lemma: Chow's Lemma: If $X$ is a scheme that is proper over a noetherian base $S$ then there exists a projective $S$-scheme $X'$ and a surjective $S$-morphism $f : X'\to X$ that ...
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15 views

Tensor product between an invertible sheaf and a constant sheaf.

This question is a natural extension this one. Consider an irreducible scheme $X$ with function field $K(X)$ and let $\mathscr L$ be an invertible sheaf over $X$. Then define the presheaf $$U\...
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1answer
22 views

“Projective” quotient of $\Bbb{Z}^2$

Consider the space of integer points $\Bbb{Z}^2=\{(x,y)|x,y\in\Bbb{Z}\}$. Consider now the equivalence relation: $$ (x,y) \sim (x',y') \quad \Leftrightarrow \quad \beta x'=\alpha x,\, \beta y'=\...
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1answer
29 views

Is the plane curve $y^3=x^4+x^3$ an irreducible algebraic affine set?

I'm dealing with the plane curve $C=\{(x,y)\in k^2:y^3=x^4+x^3\}$. I want to know if this curve is irreducible, where $k$ is a commutative field. I know this is equivalent to the ideal $\sqrt{I}$ ...
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11 views

Structures in Non-linear Sigma Model

I debated whether this belongs here, or on Physics SE, but all my questions correspond to the algebraic/complex geometry, not physics, so hopefully it's okay here. The non-linear sigma model ...
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1answer
46 views

De Rham interpretation of $H^1(R,p,\mathbb{C})$

Let $R$ be a Riemann surface of genus $g\ge 2$ and $p\in R$ a point. This is my question: Is there a way to interpret the relative cohomology group $H^1(R,p,\mathbb{C})$ as a De Rham cohomology group ...
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77 views

For which $n$ can $(a, nb, c)$ and $(b, c, d)$ be Pythagorean triples?

Fermat proved that if $(a, b, c)$ is a Pythagorean triple, then $(b, c, d)$ cannot be a Pythagorean triple. Suppose $(a, nb, c)$ form a Pythagorean triple. Can $(b, c, d)$ be a Pythagorean triple? ...
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25 views

Quick question: Curvature form of a connection on the trivial bundle

Let $L=\mathbb{R}^2\times U(1)$ be the trivial $U(1)$-bundle over $\mathbb{R}^2$. Define a connection $\nabla=d+A$ where $A=fdx+gdy$ is an $\mathbb{R}$ valued $1$-form on $L$. That is, $\nabla$ gives ...
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23 views

Sheafifying direct sum of twists

Let be $X\subseteq \mathbf{P}^r$ a smooth projective variety and let be $\mathscr E$ an invertible sheaf over $X$. Let $$M=\bigoplus_{n\geq 0} H^0(\mathscr E(n))$$ as a module over the polynomial ...
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66 views

Map of tangent spaces is the Jacobian in Algebraic Geometry

I need your collected brainpower to help me out. This is going to be long, so grab your favorite beverage and snack. I am working through Görtz and Wedhorn's "Algebraic Geometry I" and I am currently ...
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1answer
61 views

First cohomology group on a Riemann surface with all wedge products equal to zero

Sorry for the strong edit, but I realized my question had a easier formulation: Can there be a Riemann surface $X$ with the property $\sigma\wedge \tau=0$ for every $\sigma,\tau\in H^1(X,\mathbb{C})$?...
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30 views

DeRham complex of an algebraic variety; which cohomology do we use?

Let $X$ be a smooth algebraic variety of dimension $n$ over a field $k$. Let $\Omega_X$ be the sheaf of differentials (over $k$). Then we may consider the deRham complex $$\Omega_X^\bullet= \mathcal{O}...
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1answer
61 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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1answer
31 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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23 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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1answer
33 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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29 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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35 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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31 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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33 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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34 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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1answer
24 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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34 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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41 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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33 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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47 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
52 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
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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51 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 ...
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1answer
60 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
34 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
40 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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21 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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1answer
50 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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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
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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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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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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1answer
40 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
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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2answers
46 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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0answers
16 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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1answer
61 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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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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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 ...
3
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3answers
102 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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1answer
77 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 $G = Z/3Z =$ $\{1, \omega, \omega^2\}$, ...