The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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How do we know that $f(x)\in Y$?

At page 19 in this book $f:X\to Y$ is defined to be $$f(a):=(\tilde\varphi(T_1')(a),\dots,\tilde\varphi(T_n')(a)).$$ To explain the notation above, $X\subseteq \mathbb{A}^m(k)$, $Y\subseteq ...
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24 views

Injective and continuous function that is an embedding

Consider $n,d\in \mathbb N$ and $N= {n+d\choose d}-1$, then the well known $d$-uple embedding: $$\rho_d: \mathbb P^n(\mathbb C)\longrightarrow\mathbb P^N(\mathbb C)$$ is a continuous (respect to ...
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27 views

Fulton's algebraic curves question 1.36

I'm trying to solve the question 1.36 from Fulton's algebraic curves book: Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and $\dim_{\mathbb C}(\mathbb C[X,Y]/I)$ Obviously ...
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19 views

Support of the pullback module

Let $X$ be an algebraic variety, let $\Delta : \mathrm X \to \mathrm X^2$ be the diagonal embedding and let $\mathrm M$ be a quasi-coherent sheaf of modules on $\mathrm X^2$. Make the supposition ...
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47 views

What is $k(X)[Y]$ and why is it a principal ideal domain? From a proof in Fulton's Algebraic Curves

Fulton's "Algebraic Curves" says the following: Let $F$ and $G$ be polynomials belonging to $k[X,Y]$, where $k$ is a field. Let $F$ and $G$ not have a single common factor in $k[X][Y]$. Then they ...
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33 views

$SU(n)$ as a variety

Consider the algebraic group $SU(n)$ as an algebraic group scheme over $\mathbb R$. Is it birational to an affine space over $\mathbb R$?
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45 views

Singular Chain of a Hyperplane.

I refer to the definitions of Hatcher's Algebraic Topology. Is it possible to model a hyperplane $H$ (or half of it) of $\mathbb{R}^n$ with a singular chain? And if - how would its boundary look like? ...
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27 views

Residue sequence

I'm reading book Compact complex surfaces. In the first section of the second chapter they consider a curve $C$ on a surface $X$ (for simplicity I assume that $X$ and $C$ are smooth), then tensoring ...
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38 views

Geometric Interepretation of $\mathbb{G}_a$-torsors

Let's fixed a locally ringed space $(X,\mathcal{O}_X)$ (although, this should apply to any ringed topos, but I haven't thought that through). In fact, if it's helpful, you can assume that $X$ is a ...
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36 views

A couple of questions regarding algebraic sets.

I have two questions regarding algebraic sets. Let $k$ be a field. Given a finite set of points $S\subset k^n$, can we always find a set of polynomials $T\subset k[x_1,x_2,\dots,x_n]$ such that ...
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25 views

Intersection of a curve on an exceptional divisor with the exceptional divisor

Let $q: W \to X$ be a proper birational morphism of quasi-projective varieties. Let $E$ be an effective, $q$-exceptional divisor. Suppose $C \subseteq E$ is a curve, then is it true that the ...
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31 views

Show that there is $\tilde\varphi$ which makes a diagram involving reduced finitely generated $k$-algebras commutative

Let $k$ be an algebraically closed field, $X\subseteq \mathbb{A}^m(k)$, $Y\subseteq \mathbb{A}^n(k)$ affine algebraic sets and $\varphi:\Gamma(Y)\to\Gamma(X)$ be a morphism of reduced finitely ...
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55 views

Action of $\mathbb Z_2$

Is there a connection between Artin-Schreier theorem on finite groups which can be absolute Galois groups and the classification of finite groups freely acting on even-dimensional sphere? The former ...
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76 views

“Equivalent” definitions of the gluing axioms

I tried to convince myself that the two caracterizations of a presheaf that is a sheaf given in wikipedia are equivalent but I couldn't. (F presheaf and notations from wiki) Let's take a simple ...
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72 views
+50

Video lectures of algebraic geometry (Hartshorne, Shafarevich, … )

I am a commutative algebra student. I wonder if there is some video lectures of algebraic geometry courses available online for free? (I want to download) I'd like the lectures to cover ...
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34 views

Coordinate ring of the product of projective variety

Let $X\subseteq \mathbb{P}^r,Y \subseteq \mathbb{P}^s$ be two projectve varieties,what is the coordinate ring of $X\times Y$(segre embedding)?Is it true that $$S(X\times Y)=S(X)\otimes_k S(Y)?$$ I ...
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42 views

Blow-up in Projective Space: Choosing the Appropriate Chart

Consider $x^2+y+\alpha=0$ (made-up example), where $(x,y)\in\mathbb{C}^2$ and $\alpha\in\mathbb C$ is a free parameter. The equation defines a family of curves. The curves have no common points in ...
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39 views

Weak nullstellansatz in Atiyah-Macdonald 5.17

$\newcommand{\fm}{\mathfrak{m}}$ Problem 17 in the exercises after the 5th chapter of Atiyah-Macdonald is the following (with some references and hints omitted): Let $X$ be an affine algebraic ...
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32 views

A question regarding Hilbert's Nullstellensatz.

Let $k$ be an algebraically closed field, and $a$ an ideal of the polynomial ring $k[x_1,x_2,\dots,x_n]$. The strong form of Hilbert's Nullstellensatz says that $I(Z(a))=\sqrt{a}$. Note:- Initially, ...
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58 views

Decomposition of an algebraic variety into irreducible components

I'm studying the Fulton's algebraic curves book and I have the following doubts in the end of the page 9: I didn't understand why the following equations hold: $$I\left(\bigcup_i ...
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27 views

Hartshorne Example I.5.6.3

This question is concerned with Example I.5.6.3 in Hartshorne. Let $g, h$ be elements of $k[[x,y]]$ of the form $g = y+x+g_2+g_3+\cdots, h = y-x + h_2+h_3+\cdots$ where $g_i,h_i$ are homogeneous ...
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26 views

function degree

I have the function $ I:{ \mathbb{P}^{2} - \{[1:0:0]\} - \{[x_{0}:x_{1}:x_{2}] | x_{0}=0\} }\longrightarrow \mathbb{P}^{2} - \{[1:0:0]\} - \{[x_{0}:x_{1}:x_{2}] | x_{0}=0\} $, where ...
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31 views

Lines In the Complex Proyective Plane

The question is In how many points a line in CP^n intersects CP^2?. By a line in CP, I mean a copy from CP^1. I have tried with a sytem of equations, (Because a line in CP^n is the zero locus of a ...
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41 views

Hartshorne Theorem I.5.3

This question concerns a reduction argument that occurs in the proof of Theorem I.5.3 in Hartshorne. In particular, let $Y$ be an affine variety of dimension $r$ in $\mathbb{A}^n$. Then by (4.9) $Y$ ...
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30 views

solving system of equations(nonlinear)

I am trying to solve the following system of equations: $$\frac{kq^2}{d}=mg(L-L\cos(t))+\frac{kq^2}{r}$$ $$\sin(t)=\frac{x}{L}$$ $$r^2=(L-L\cos(t))^2+(x+d)^2$$ The parameters are: $k,L,d,q,m,g$ The ...
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17 views

Is a projective system of finite etale covers of a scheme S the same as its limit?

Fix a scheme $S$, and let $\text{FEt}_S$ be the category of finite etale covers of $S$. My question is, is Pro-$\text{FEt}_S$ equivalent to the full subcategory of $\text{Sch}/S$ consisting of ...
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1answer
21 views

Is the base extension to K of an irreducible nonsingular projective variety over k irreducible?

Suppose $X$ is an irreducible nonsingular projective variety over a field $k$ (not necessarily algebraically closed) Let $K$ be a field extension of $k$ ( If $K/k$ is not algebraic, we can assume ...
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48 views

Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
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21 views

Where to learn about the Chow scheme and the Hilbert-Chow morphism?

I would like to learn something about the Chow scheme of cycles on an algebraic variety. I am not after an abstract treatment of the moduli problem in full generality, actually I would be happy with a ...
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51 views

Connection between differential form of a manifold and Sheaf of relative differential of a map of schemes.

I was wondering whether there is a connection between the differential form of a manifold and Sheaf of Relative differential of a Scheme map. Definitions: Differential form on a manifold M is a map ...
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1answer
38 views

Invertible sheaves on affine varieties

Let $X:=\rm{Spec}(A)$ be an integral, noetherian, affine variety, and let $L$ be an invertible sheaf on $X$, I try to find an example where $L$ is not isomorphic to the structure sheaf of $X$. In the ...
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23 views

Reference request on numeric semigroups

I watched some talks about numeric semigroups, and thei relation whti algebraic geometry (such as Weierstrass semigroup of a curve), and I'm interested in take a deeper look in this topic, can anyone ...
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32 views

Projective varieties and irreducibility

The "modern"(schematic) definition of a projective variety is the following: Let $k$ be an algebraically closed field. A projective variety over $k$ is a closed subscheme of $\mathbb ...
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1answer
27 views

Is a ruled surface of degree>2 always singular?

Let $X=\mathbb{C}\mathbb{P}^3$ and let $V\subset X$ be a closed algebraic sub variety. By V-is ruled, I mean that for every point in $V$ there is a line passing through it which also lies in V. ...
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34 views

Tensor product with an L on top

Looking at the definition of the Kunneth morphism in SGA4, XVII, 5.4.1.4, there is the notation $$Rf_*K \overset{\mathbb{L}}{\boxtimes}_{\mathcal{A}_0} Rg_* L \rightarrow Rh_*(K ...
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43 views

Are noetherian hypotheses necessary for the theory of the etale fundamental group?

The etale fundamental group, as explained in SGA 1 Expose 5 and various other notes I've read, always makes the assumption that the scheme $S$ (for which one intends to construct a fundamental group), ...
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43 views

Homotopy groups relating to toric varieties

It is known that the toric variety $X_\Sigma$ of a simplicial fan $\Sigma$ can be constructed as a quotient $$X_\Sigma = \bigl(\mathbb C^N \setminus V(B)\bigr)/G.$$ Here $N$ is the number of rays, ...
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2answers
44 views

Some questions on Hartshorne I.7: intersections in projective space

I am reading I.7 of Hartshorne, and here are some questions I don't understand. 1) Prop. 7.4. Let $M$ be a finitely generated graded module over a noetherian graded ring $S$. Then there exists a ...
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39 views

Resolving the Base-points through Blow-ups

This is related to a question I asked earlier: Link So, the Hesse pencil is given by $\lambda (x^3+y^3+z^3)+\mu xyz=0$, where $[\lambda,\mu]\in\mathbb{CP^1}$ and $[x,y,z]\in\mathbb{CP^2}$. I can ...
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21 views

scheme-theoretic image behaves nicely with composition, base change?

Scheme-theoretic image is still somewhat of a mystery to me, and I wasn't able to work out proofs of either of the following two statements that seem plausible to me: If $X\to Y\to Z$ is a map of ...
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69 views

Only $f^\sharp_x$ makes the diagram commutative

By Algebraic Geometry I from Görtz, Wedhorn page 60 $f^\sharp_x$ is the unique ring homomorphism which makes the diagram $A\to B \to B_{p_x}$, $A\to A_{p_{f(x)}}\to B_{p_x}$ commutative. The first ...
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58 views

List of exercises and examples to see the geometry behind algebraic geometry

What exercises should one solve (understanding proofs included) to gain an intuition for algebraic geometry? What are examples of (not too hard) problems that algebraic geometry handles easier than ...
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111 views

What are local homomorphisms, geometrically?

For want of a better name, let us say that a ring homomorphism $f : A \to B$ is local if it (preserves and) reflects invertibility, i.e. $f (a)$ is invertible in $B$ (if and) only if $a$ is invertible ...
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40 views

Hesse's Pencil: Base Points and Resolution of Singularities through Blow-ups

A snippet of the definition given on wikipedia (full link: here) The Hesse pencil is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the ...
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55 views

do the homomorphisms between two group schemes form a sheaf in the (whatever)-topology?

By this I mean: Suppose you have two group schemes $G,H$ over a scheme $S$. Then you have a presheaf on the category $\text{Sch}/S$ sending $$(T\rightarrow S)\mapsto\text{Hom}_T(G_T,H_T)$$ which is a ...
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1answer
60 views

Asterisk Notation

I am trying to read through the Cech Cohomology section in these notes: http://pub.math.leidenuniv.nl/~edixhovensj/teaching/2011-2012/AAG/lecture_14.pdf I would be really grateful if someone could ...
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2answers
115 views

roots of a polynomial inside a circle

I am asked to show that for $n$ larger or equal to $2,$ the roots of $1 + z + z^{n}$ lie inside the circle $\|z\| = 1 + \frac{1}{n-1}$ Attempt1: Induction for the case $n = 2,$ the roots of $1 + z + ...
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65 views

Blow-Up over a Field

I want to prove that a function $\pi : \mathbb{C}_{*}^{n}\mapsto \mathbb{C}^{n}$ is bijective. Where $\mathbb{C}_{*}^{n}$ is the explosion of $\mathbb{C}^{n}$ and is defined as $\mathbb{C}_{*}^{n}:= ...
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1answer
50 views

Base-points and invertible sheaves

Once again I am confused after thinking too much about something I thought I already understood... Let $\mathcal{L}$ be an invertible sheaf on a smooth projective curve $X$ such that $\deg ...
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1answer
34 views

Matrix notation of an ellipse.

When I was reading a paper related to computer vision, I came across the following notation, where an ellipse is represented by the equation $\mathbf{x}^TM\mathbf{x} = 1$, where the ellipse parameter ...