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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153 views

Proving algebraically that $\mathbb RP ^3\cong SO(3,\mathbb R)$

I am giving a simple introductory course on algebraic geometry and I plan to mention that $$\mathbb RP ^3\cong SO(3,\mathbb R).$$ I know a rather simple proof of this using the fact that $\mathbb ...
5
votes
2answers
158 views

On a *ringed space*, show that the non vanishing set of $f$ is open, and that it is invertible there

This is an exercise of Ravi Vakil that I solved by a very trivial argument without using the hint. For this reason I'm worried that I might have missed something. If $f$ is a function on a locally ...
5
votes
2answers
208 views

Function field of an affine hypersurface

I am reading Hulek' Elementary Algebraic Geometry, p.103 Let $V$ be an irreducible affine hypersurface, say $V=V(f)\subset\Bbb{A}^n$. Then the coordinate ring is by definition ...
2
votes
2answers
272 views

Source: Coherent locally free sheaves and projective modules

What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?
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2answers
173 views

Parametrization of the line in the projective space

Let $L=aX+bY+cZ$ be a line in the projective space, the book I'm using states that every such line has the following parametrization: $$\varphi:\mathbb P^1\to L, \ (t:s)\mapsto ...
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223 views

is the image of a polynomial map contractible?

I asked this in MSE http://math.stackexchange.com/questions/643348/is-the-image-of-a-polynomial-map-contractible and got no response. Either it's a silly question or I posted it under the wrong ...
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2answers
54 views

Calculating a circle's radius from one point and two circles on it's circumference

Suppose that there are four points $A, B, C, D$. A circle of radius $r_A$ surrounds point $A$, a circle of radius $r_C$ surrounds point $C$, and a circle of radius $|DB|$ surrounds point $D$. ...
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1answer
40 views

Ring of rational functions ideal generators

There is an affine variety $X\subset \mathbb{A}^n$ with its ring of rational functions which is the quotient ring of $\mathbb{k}[X]$ (each $f\in \mathbb{k}(X)$ has a form $\frac{p}{q}$ where $q$ does ...
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1answer
91 views

Scheme theoretic image of a base change of a morphism of schemes

Let $f\colon X \rightarrow Y$, $g\colon Y' \rightarrow Y$, be two morphisms of schemes. Let $X' = X\times_Y Y'$, and let $f'\colon X' \rightarrow Y'$ be the projection. We are interested in the ...
2
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2answers
146 views

When do we have $\operatorname{depth}_{A} B = \operatorname{depth}_B B$?

Let $(A,\mathfrak{m}) \to (B,\mathfrak{n})$ be a local homomorphism with $A$ a regular local ring. Assume further that this ring map is finite. How can we prove that $\operatorname{depth}_B B = ...
2
votes
1answer
69 views

Finding irreducible components over $\mathbb Q$ and $\mathbb C$

I want to find the irreducible components over $\mathbb Q$ and $\mathbb C$ for some curves, namely, (a) $Y^2=X^5$ (b) $Y^2=X^3+1$ Intuitively, it seems to be that if the equation is not ...
3
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1answer
152 views

base change of a reduced scheme over $\mathbb{Z}$

I'm reading through the stacks project and came across a lemma along these lines: Let X be a scheme over a perfect field k. Then, $X$ is reduced implies $X$ is geometrically reduced. here is my ...
15
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1answer
289 views

What is the connection between Weil's character bound and Riemann Hypothesis over finite fields

Weil's character bound states that: Let $\mathbb{F}_{q}$ be a finite field of size $q$. Let $\chi$ be a multiplicative character of order $m$. Let $f(x)$ be a polynomial of degree $d$ such that $f(x) ...
3
votes
2answers
258 views

Example of an affine scheme where closed points aren't dense.

I'm looking for an example of an affine scheme where closed points aren't dense. It's easy to show (using Hilbert's Nullstellensatz) that if $A$ is a finitely generated algebra over a field, then the ...
8
votes
1answer
120 views

Inhomogeneous polynomial and points at infinity

Let $f=X^2-Y$ be a polynomial in $k[X,Y]$, so $V(Z)$ is a parabola: $V(f)$: According to Bézout theorem the $y$-axis has to intersect the parabola two times. We know the y-axis meets the ...
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0answers
65 views

How to apply “proper base change” here

Reading a book about curves I encoutered the following claim, which I don't understand. Let $X$ be a smooth projective curve, and $\nu:X\times \mathrm{Pic}(X)\to \mathrm{Pic}(X)$. Pick a universal ...
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0answers
46 views

G action on schemes

Let $X$ be a smooth projective variety over $k$ and $G$ a finite group such that $\mathrm{char}(k)$ does not devide the order of $G$. Then for some $G$-equivariant coherent sheaves $\mathcal{F}$ and ...
3
votes
1answer
271 views

Projections are finite morphisms

Let $X$ be a variety in $\Bbb{P}^n$. I would like to see as simply as possible why the projection of $X$ from a point is a finite map. Suppose $p=(1:0:\ldots:0)\notin X$ and let ...
4
votes
1answer
53 views

Understanding the Affine Case of a Stacky Result

I'm going through Vistoli's sections of FGA Explained to begin to understand stacks. It is well-known and proven in the text that the fibered category $QCoh$ of quasi-coherent sheaves is a stack in ...
6
votes
1answer
1k views

Is there any English version of Récoltes et Semailles?

I felt like my question isn't appropriate for MO, so I though maybe I should post it here. I want to read Alexander Grothendieck's "Récoltes et Semailles", but I don't know any French. I can easily ...
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1answer
63 views

Are holomorphic maps regular maps of varieties?

Is a holomorphic map of complex algebraic varieties always a regular map?
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81 views

Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial $$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n} $$ After the linear change of ...
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0answers
58 views

canonical bundle

We are given smooth projective varieties $X$ and $Y$ over $k$. Suppoese $X\simeq Y$. Do we have $\omega_X\simeq \omega_Y$? If this is the case, then for $\pi_1:\mathbb{P}(\mathcal{E})\rightarrow X$ ...
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2answers
630 views

Is skyscraper sheaf quasi-coherent?

Suppose $\mathcal{F}$ is a skyscraper sheaf supported on $\bar{\{\mathfrak{p}\}}$, the stalk is $M$, What is its global section over $\operatorname{Spec} A$? We need to find a module $N$ such that ...
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votes
1answer
81 views

Inverse question: Recognizing when a phenomenon behaves like an algebra

I might be phrasing this question incorrectly, but my students asked me about it and I did not have a good answer. I am in statistics and I look at all sorts of social data, network data, climate ...
5
votes
2answers
132 views

Local cohomology with respect to a point. (Hartshorne III Ex 2.5)

I'm trying to do Hartshorne's exercises on local cohomology at the moment and seem to be stuck in Exercise III 2.5. The problem goes as follows: $X$ is supposed to be a Zariski space (i.e a ...
2
votes
1answer
104 views

The $\mathbb C((z))$-rational points of a complex semi-simple group $G$

By definition, if $R$ is a $\mathbb C$-algebra and $G$ is a $\mathbb C$-scheme then the set of $R$-valued points on $G$ is $G(R)=\hom_{\text{Sch}_{\mathbb C}}(\operatorname{Spec} R, G)$ In Ginzburg's ...
8
votes
1answer
83 views

Disjoint standard open sets in Spec(R)

The following appeared as a homework problem last semester in Johan de Jong's algebraic geometry course at Columbia (http://www.math.columbia.edu/~dejong/schemes.html), described as "a bit of a ...
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1answer
38 views

Group law on invertible fractional ideals on a scheme

I have some questions about the group of invertible fractional ideals on a scheme $X$. What is the group law ? In Görtz-Wedhorn book (paragraph 11.12), it is written that product of two invertible ...
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1answer
309 views

Generalizations of Hilbert's Syzygy theorem

Hilbert's Syzygy theorem states that a minimal free resolution of a finitely generated graded module over a (standard graded) polynomial ring in $n$ variables $k[x_1, \ldots, x_n]$ does not have more ...
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1answer
123 views

A Question on “flatness is preserved by base change”

Let $f:X\to Y$ be a morphism, $\mathcal{F}$ be an $O_X$ module which is flat over $Y$, let $g:Y'\to Y$ be any morphism. Let $X'=X\times_YY'$, let $f':X'\to Y'$ the second projection, and ...
3
votes
1answer
125 views

Example of a curve of genus $4$

I'd like to put my hands on some polynomial defining a curve of genus $4$, living in the plane or in the 3D space. Do you know about any? Is there any procedure to build one? The best would be one ...
5
votes
1answer
221 views

Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
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348 views

Position and nature of singularities of an algebraic function (Ahlfors)

I want to solve the following exercise, from Ahlfors' Complex Analysis text, page 306: Determine the position and nature of the singularities of the algebraic function defined by $w^3-3wz+2z^3=0.$ ...
3
votes
1answer
204 views

localization in algebraic geometry

It is often asserted in commutative algebra texts that localization is important in algebraic geometry. I would appreciate some precise examples which show the utility of the concept in this context. ...
4
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1answer
266 views

Some questions on Hartshorne III Ex 6.8

I have been looking at Hartshorne III exercise 6.8 for nearly a week now and I don't seem to have a clue as to how to do it. In particular, I am stuck on part (a) which boils down to showing the ...
4
votes
2answers
115 views

Krull-Schmidt-Remak for vector bundles

I'm reading Nori's paper The fundamental group scheme, and I have some problems in certain passages of the proofs. This one is from chapter 1, 2.3. Let $X$ be a complete connected reduced ...
4
votes
1answer
264 views

Cotangent bundle of complex manifold is Calabi-Yau manifold

We say that a complex manifold $M$ is Calabi-Yau if the canonical bunlde is trivial $K_M=0$. How can we prove that the total space of the cotangent bundle of a compact complex manifold $N$ is ...
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0answers
32 views

Relation of Between inseparable morphisms and tangent lines

$X$ be a projective curve of $\mathbb{P}^n$, $P$ is not $X$ and let $f:X\rightarrow \mathbb{P}^{n-1}$ be the projection from $P$. When I read Hartshorne book, I see that if $f$ is insepable (i.e the ...
4
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1answer
224 views

Line bundle corresponding to the Segre embedding

I am trying to understand the theorem that characterizes morphisms to projective space as equivalent to the data of a line bundle together with global sections generating it. I tried to find the ...
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1answer
41 views

Proposition 4.3.9 in Liu: flatness by domination

Proposition 4.3.9 in Liu says: Let $Y$ be a Dedeking scheme. Let $f:X\to Y$ be a morphism with $X$ reduced. Then $f$ is flat if and only if every irreducible component of $X$ dominates $Y$. I don't ...
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0answers
187 views

Skyscrapers sheaf's global sections

I'm reading a book written by Serre and, even though he's one of the best math writer ever, there's a step I don't understand. This may imply that I'm one of the worst math reader ever! ...
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1answer
56 views

How to determine the group structure of $E(\mathbb{R})$ for an elliptic curve $E/\mathbb{R}$

Using Weierstrass' $\wp$ function it can be proved that the group of complex points on an elliptic curve $E /\mathbb{C}: y^2 = x^3 + ax + b$ satisfies $E(\mathbb{C}) \cong \mathbb{R}/\mathbb{Z} \oplus ...
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62 views

Yoneda's lemma to prove $f^*(\tilde M) \cong \widetilde{B\otimes _A M}$

Let $M$ be an $A$-module. and let $\phi : A \rightarrow B$ be a ring homomorphism. Let $f$ be the corresponding morphism of scheme from $SpecB$ to $Spec A$. Then prove that $f^*(\tilde M) \cong ...
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65 views

Computing an Euler characteristic

Let $X$ be an abelian variety of dimension $3$. I wish to compute the Euler characteristic of a certain subscheme $V$ of the Hilbert scheme $H=\textrm{Hilb}^3X$. Definition of $V\subset H$: it ...
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1answer
68 views

Ampleness, Nakai's criterion and pullback

In the book I'm reading ( Geometry of Algebraic Curves ), at some point (page $310$) they make the following claim: One can use Nakai's criterion to establish the general fact that if $f:X\to Y$ ...
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89 views

Principal Divisors of rational function

Let $X$ be a nonsingular curve. Let $k$ be algebraically closed field. If $f$ is a rational function on $X,$ then the principal divisors of $f$ is defined by $(f) = \sum_{P\in X}\nu_{P}(f)P,$ where ...
3
votes
1answer
470 views

pull-back and push-forward of quasi-coherent sheaves on affine schemes

Let $f:Y\to X$ be a map of affine schemes, where $X=\text{Spec}A$ and $Y=\text{Spec}B$. Let $M,N$ be modules over $A$ and $B$, respectively. I know the following three facts: The functors $f^{*}$ ...
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4answers
411 views

Hartshorne's Exercise II.5.1 - Projection formula

I'm trying to solve Exercise 5.1 of Chapter II of Hartshorne - Algebraic Geometry. I'm fine with the first $3$ parts, but I'm having troubles with the very last part, which asks to prove the ...
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1answer
46 views

Rational locus of a function defined on $x^2+x^3=y^2$

We have a curve $X$ on $\mathbb{A}^2$ given by $y^2=x^2+x^3$. Consider the rational function $f$ on $X$ which maps $(x,y)\in X$ to $\frac{y}{x}$. There is a nice geometric interpretation of $f$: if we ...