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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Factoring a birational morphism through blowup

Let $X,Y$ be smooth, proper varieties, and $f: X \to Y$ be a proper birational morphism. Suppose $E$ is a smooth, irreducible exceptional divisor, with the image $f(E)$ also smooth. Let $I$ be the ...
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35 views

finite group actions

Let $X$ be a smooth quasi-projective and separated $k$-scheme and $G$ a finite group acting on $X$. Suppose $\mathrm{char}(k)$ does not divide the group order. Then there is the quotient stack ...
6
votes
1answer
53 views

Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
2
votes
1answer
94 views

Bezout's bound and resultants - reference request

In Terry Tao's blog post about Bezout's inequality, he writes: In our notation*, this theorem states the following: Theorem 1 (Bezout’s theorem) Let $d=m=2$. If $V$ is finite, then it has ...
1
vote
1answer
152 views

Projection of a hypersurface from a point

Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X_n=0\}$ and a hypersurface $X$. We want ...
3
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0answers
57 views

Example for divisors, line bundles and meromorphic functions on $\mathbb{CP}^2$

I have been studying divisors using Griffiths/Harris (chapter 1.1) as well as Huybrechts (chapter 2.3). However, I cannot seem to find any very easy worked examples - i.e. $\mathbb{CP}^1$ or ...
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1answer
44 views

Generating set of the algebra invariants of finite group.

Let a finite group $G$ acts on a complex vector space $V$ and let $\mathbb{C}[V]^G$ be corresponding algebra of polynomial invariants. Let $f_1,f_2,\ldots,f_m$ be a generating set of this algebra of ...
2
votes
1answer
50 views

$\mathbb C$-isomorphism between two $\mathbb C$-schemes.

Consider a field automorphism $\sigma\in\textrm{Aut}(\mathbb C)$, and moreover consider the $\mathbb C$-scheme $p:\mathbb P^1_{\mathbb C}\longrightarrow\textrm{Spec}\,{\mathbb C}$ where $\mathbb ...
0
votes
1answer
43 views

The ring of fractions $K(x)$ is the field generated by $K$ and $x$.

I would like to show that the ring of fractions $K(x)$ of $K[x]$ in an extension $L$, where $K\subset L$ fields, is the field generated by $K$ and $x$ (let's call it by $\tilde{K(x)}$). I know just ...
0
votes
2answers
19 views

Rational functions are decomposed in polynomial products

I'm trying to understand why this is true: Since $K(x)$ is a field, $K(x)$ is an UFD, then $K(x)$ can be written uniquely as products of irreducible elements of $K(x)$. I didn't understand why ...
3
votes
1answer
36 views

Characterization of the transcendentals over a field

I'm studying Algebraic Function Fields and Codes book from Henning Stichtenoth and I didn't understand this remark in the first page: I couldn't solve any part of the equivalence, I think maybe ...
0
votes
0answers
19 views

projective change of coordinates and tangent line

Let $C/k$ be a projective algebraic curve given by a polynomial $F \in k[X,Y,Z]$. If $L$ is tangent to $C$ at a point $P$ does $L$ remain tangent to $C$ at a point $P'$ after a projective change of ...
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0answers
27 views

Morphisms of quasi-projective varieties

Let $Y\subseteq \mathbb{P}^n(k)$ be a quasi-projective variety. By Görtz, Wedhorn (page 32, Proposition 1.65) in order to show that $$h:Y\to \mathbb{P}^m(k), y\mapsto (f_0(y):\dots :f_m(y))$$ is a ...
3
votes
1answer
28 views

Intersections on General Nonsingular Projective Varieties

Let $X$ be a nonsingular, integral projective variety of dimension at least 2 over $k$ algebraically closed. Let $Y$ and $Z$ be two codimension 1 subschemes (effective Weil divisors) of $X$. Must they ...
2
votes
0answers
23 views

Analytic cohomology on Zariski site vs analytic cohomology on analytic site

If I have an affine algebraic complex manifold (in fact it is Stein), what is known relating the cohomology of analytic sheaves using only Zariski opens vs the cohomology of analytic sheaves using the ...
-1
votes
1answer
29 views

Reference request: About Weil book

In "Standard conjectures on algebraic cycles" of Grothendieck and "Algebraic cycles and the Weil conjectures" of Kleiman they say in their references: A. Weil: Variétés Kählériennes, Hermann, ...
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1answer
45 views

$a,b$ integral $\implies$ $a+b$ integral

I'm sure it's just silly thing. I'm reading Fulton's algebraic curves book and I don't understand this phrase of this proof: I didn't understand why according to the proposition we have $a\pm b,ab$ ...
2
votes
0answers
47 views

About the isomorphism of two schemes.

Let $C$ and $B$ be two graded $A$-algebras, where $A$ is a commutative ring with unity. Look at the following lemma from Liu's book: Now suppose that $\varphi$ is an isomorphism of graded ...
0
votes
0answers
101 views

Blow up of base locus of a pencil and line bundles

This is related to a previous question of mine: Is the universal hyperplane section the blowup of the baselocus? Let $X$ be a variety, $L$ a line bundle on it. Take $V < H^0(X,L)$ to be a linear ...
3
votes
1answer
36 views

Can you have a nontrivial automorphism of an elliptic curve $E/S$ which when restricted to a geometric fiber is the identity?

Ie, let $E/S$ be an elliptic curve over some scheme $S$. Is it possible to have an automorphism $\alpha$ of $E$ over $S$ such that for some geometric point $s\in S$ its pullback to $E_s$ is the ...
7
votes
3answers
139 views

For which $n$ is $\mathbb{A}^n\setminus \{0\}$ affine?

For which $n$ is $\mathbb{A}^n(k)\setminus \{0\}$ an affine variety? I think for $n=0$ and $n=1$ it is. For $n>1$ probably not, but I don't have a proof. $n=0$: Take the ideal $\mathfrak{a}:=(1)$ ...
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2answers
84 views

What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...
1
vote
1answer
86 views

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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votes
0answers
43 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 ...
0
votes
2answers
74 views

Dimension of $\dim_{\mathbb C}\mathbb C[X,Y]/I(Y^2-X^2,Y^2+X^2)$

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 ...
0
votes
0answers
29 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 ...
2
votes
1answer
55 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 ...
0
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0answers
38 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 $\mathbb R$?
0
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1answer
57 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? ...
3
votes
0answers
54 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 ...
1
vote
1answer
46 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 ...
0
votes
1answer
37 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 ...
1
vote
1answer
34 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 ...
1
vote
1answer
57 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 ...
1
vote
2answers
101 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 ...
7
votes
3answers
303 views

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'd like the lectures to cover main topics of the books ...
3
votes
1answer
54 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 ...
2
votes
1answer
51 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 ...
0
votes
1answer
44 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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1answer
38 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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1answer
65 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 ...
2
votes
1answer
30 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 ...
0
votes
0answers
27 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 ...
1
vote
1answer
35 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 ...
0
votes
1answer
46 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) $Y$ ...
1
vote
1answer
34 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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votes
0answers
22 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 ...
1
vote
1answer
23 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 ...
3
votes
1answer
54 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 ...
1
vote
0answers
24 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 ...