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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3
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1answer
27 views

Formal schemes vs formal power series

Take $X = \mathbb{A}^1$ and $Y = \{0\}$. I want to take the formal group scheme at $Y \subset X$. This is a locally ringed space, $(Y, \mathcal{O}_{ \hat{X}})$ where $\mathcal{O}_{\hat{X}}$ is the ...
0
votes
0answers
15 views

An isogeny from a split algebraic torus

Suppose that there is an isogeny (in the category of commutative algebraic groups) from a split algebraic torus to a semi-abelian variety. Does it follows that this semi-abelian variety is also an ...
9
votes
1answer
121 views

Why is there no theory of $G$-ic varieties, for linear algebraic groups $G$?

A toric variety is an algebraic variety $X$ with an embedding $T \hookrightarrow X$ of an algebraic torus $T$ as a dense open set, such that $T$ acts on $X$ and the embedding is equivariant. It ...
0
votes
2answers
40 views

Fermat's Curve is not rational (Perrin's “Algebraic Geometry - An Introduction”)

Below is the proof that curve given by $x^n+y^n=1$ for $n\geq 3$, over field of characteristic that does not divide $n$, has no rational parametrization, from Intrduction of Perrin's "Algebraic ...
0
votes
1answer
21 views

Decomposition into irreducible algebraic sets

I am facing following problem and would really appreciate anyone's help: I am to find decomposition of $V(x^2 - y^4, x^3 - xy^2 + x^2y^2 -y^4)$ into irreducible algebraic sets in $A^2( \mathbb{C})$. ...
2
votes
2answers
104 views

Why the unitary group is not a complex algebraic variety?

The question comes from Exercise 1.1.2 of the book "An Invitation to Algebraic Geometry". By definition the unitary group U(n) is the group of all complex matrix that satisfies $U^*U=I$. I know that ...
3
votes
1answer
8 views

Containment of two varieties with a lot of intersection

Given a projective variety $X\subset \mathbb P^n$ and a curve $C\subset \mathbb P^n$, when can I conclude that $C\subset X$, from the fact that $C$ and $X$ have 'many' points in common. I.e., is there ...
3
votes
1answer
55 views

When do functions turn a space into a locally ringed space?

Let $X$ be a topological space, and consider for each open set $U \subseteq X$ a set $F_U$ of functions $U \to k$ into some fixed field $k$. Let $\mathcal{O}$ be the sheaf of $k$-algebras induced by ...
1
vote
1answer
26 views

Generators of ideal of coordinate axes in A^3

So I know that the algebraic set $X$ equal to the union of the three coordinate axes in $\mathbb{A}^3$ is $I(X) = (xy, yz, xz)$ and that this is the fewest number of generators. But it seems that the ...
6
votes
0answers
52 views

“First” results in algebraic geometry where schemes are needed

I'm interested where in the study of algebraic geometry, one really needs the full theory of schemes for the first time? Are there any results about varieties where using Hartshorne Ch1 type of ...
1
vote
1answer
66 views

Intersection of two polynomial ideals

In the $4$-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$ X'=\{x=y=0\} $$ and $$ X''=\{z=x-t=0\} $$ (I'm working on a algebraically ...
0
votes
0answers
24 views

$\text{div} (\mathcal{L},s) = 0$

I am reading a proposition in Prof. Vakil's notes where he shows that if $X$ is a normal and Noetherian scheme, then $\text{div}$ is injective. He opens by saying that if $\text{div} (\mathcal{L},s) = ...
1
vote
1answer
46 views

degree of morphism of schemes

Let $\phi: Y \to X$ be a finite etale morphism of proper smooth connected schemes over a field $K$ and suppose that the induced morphism $\phi: \overline{Y} \to \overline{X}$ has degree $n$, where ...
1
vote
2answers
54 views

a question regarding Klaus Hulek algebraic geometry

sorry for uploading weird angle pictures..but no other concise ways I can't think of.. this is the p.49 of Klaus Hulek elementary algebraic geoemtry on the last paragraph it says that K[V] is not a ...
0
votes
0answers
29 views

Theorem weak Hilbert

If $ I $ is a proper ideal of $ K [x_1, \ldots, x_n] = F $ then $ V (I) \neq \emptyset$. dm: I want to use that $ J = \langle x_1-a_1, \ldots, a_n x_n \rangle $ is a maximal ideal of $ F $. If I ...
1
vote
0answers
52 views

Does such a polynomial map always exist?

First question: given $$\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}=\frac{P_3(f(x_1,x_2,\dots,x_n))}{P_4(f(x_1,x_2,\dots,x_n))}|\det (J(f(x_1,x_2,\dots,x_n)))|$$ where $P_i$ is ...
3
votes
0answers
34 views

trivialising cover for etale morphisms

Let $f:Y \to X$ be a finite etale morphism of smooth and proper schemes over a field $k$ (not necessarily sep closed). Is there a geometrically connected etale cover $\{U_i\}$ of $X$ which ...
2
votes
1answer
50 views

What is the importance of Jacobian Conjecture and any progress on it?

What is the importance of Jacobian Conjecture?Are there any important central problem with the conjecture as precondition? and any progress on it?
0
votes
2answers
73 views

Algebraic variety in $\mathbb{C^{2}}$

would somebody algebraic-geometry-savvy please help me with this problem, as I am pretty new to algebraic geometry: I have to find smallest algebraic variety (irreducible algebraic set) in ...
0
votes
1answer
17 views

Plotting Particular Conic Section

How would I plot $-2x^2 -2y^2 = 1$ on the x-y plane ? I believe it is an ellipse, since the coefficients have the same sign, I just don't know what the major and minor axes would be nor how to plot.
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0answers
35 views

Counting the Number of Points in an Algebraic Variety

How can we count the number of points in $$S = \{(x,y) \in \mathbb{Z_m}^2: x^2+ky^2 = c\}$$ where $k,c$ are some positive integers?
0
votes
2answers
28 views

Curve in union of hyperplanes

If a smooth curve $\gamma: [0,T] \to \mathbb R^n$ is contained in the union of hyperplanes $$ \bigcup_{i=1, \dots, N} H_i$$ does it then follow that one can always find time intervals $[t_0, t_1]$ ...
2
votes
2answers
80 views

Discrete set of zeroes of polynomials must be finite?

Let $F:\mathbb C^n\to\mathbb C^n$ be a polynomial mapping (i.e. $n$ polynomials in $n$ variables). Suppose that $Z = \left\{z \in \mathbb C^n : F(z) = 0\right\}$ is a discrete set (all points are ...
2
votes
1answer
27 views

first chern class of cotangent bundle

If $X$ is a smooth variety, how do we see that $c_1(T_X^*)$ is the canonical divisor? I can see this for the projective space: if $X=\mathbb{P}^n$, then the Euler exact sequence $$ 0\to O\to ...
2
votes
1answer
31 views

Equality of ideals and their vareties.

Let $I_1 $,$I_2 $ $\in \mathbb{C}[x_1,x_2,...,x_n] $ be two polynomial ideals. If their affine varieties, $\mathbb{V}(I_1)=\mathbb{V}(I_2)$ are equal then is $I_1=I_2$ always?
2
votes
0answers
76 views

The greatest common divisor of homogeneous polynomials

Let a matrix $$M=\begin{pmatrix} a_{01}&a_{02}&a_{03}\\a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}$$ with $a_{ij}\in k[x,y,z]$ ...
2
votes
1answer
41 views

Solutions to a system of equations

I know that Bezout's theorem says that if you take two plane curves, then their maximal number of intersection points is the product of their degrees. However, assume that I have two irreducible ...
1
vote
0answers
29 views

intuition for the normal bundle to a divisor

I understand that if $i:D\hookrightarrow \mathbb{P}^n$ is a divisor of degree $d$, then the normal bundle to $D$ is $O_D(d)=i^*(O(d))$. The only way I know to prove this is through the conormal ...
3
votes
0answers
40 views

Dimension of a topological space [duplicate]

I want to prove this fact that if X is a topological space which is covered by a family of open subsets {U_i} than dimX=supdimU_i One direction I can see that the RHS is less than or equal to the ...
0
votes
1answer
54 views

Question regarding Vakil's algebraic geometry notes

Exercise 1.3 D of Vakil's lecture notes on algebraic geometry asks: "Verify that $A \to S^{−1}A$ satisfies the following universal property: $S^{−1}A$ is initial among $A$-algebras $B$ where every ...
1
vote
0answers
48 views

Possible mistake in Gortz-Wedhorn's algebraic geometry book

I'm trying to solve exercise 2.14c in Gortz-Wedhorn's book on algebraic geometry, and it looks to me like it's wrong. Here's the statement. Let $X$ be a topological space and $i:Z \rightarrow X$ the ...
0
votes
1answer
43 views

Is it true that ${\mathbb P}^1_{(1,2)} \cong {\mathbb P}^1$?

In a class on Algebraic geometry, we learnt the following - ${\mathbb P}^1_{(1,2)} \cong {\mathbb P}^1$ over the field ${\mathbb C}$. I'm not sure I followed the entire argument exactly. I'll ...
1
vote
1answer
75 views

Fibres of the map $Spec\mathbb{Z}[x] \rightarrow Spec\mathbb{Z}$

I am trying to understand what is a spectrum of the ring $\mathbb{Z}[x]$. I have read Spectrum of $\mathbb{Z}[x]$ but because of my very restricted knowledge of schemes I do not understand the ...
1
vote
3answers
71 views

Examples of smooth curves of genus $0$ and degree $d>2$.

Can we provide a source of explicit examples ? The degree assumption $d>2$ means that I would like to see examples which are not conics.
3
votes
1answer
70 views

Plane curves isomorphic to the affine line

Let $C$ be a plane curve parametrized by $x=f(t),y=g(t)$ where $f(t),g(t)\in k[t]$. We can easily see that the coordinate ring of $C$ is isomorphic to $k[f(t),g(t)]\subset k[t]$. So $C$ is isomorphic ...
1
vote
0answers
31 views

A question about hyperelliptic curve

This question is from the Qing Liu's book Algebraic Geometry and Arithmetic Curves 7.4.10 Let P(t) $\in$ k[t] be a seperated polynomial of even degree $\geq$ 2 over an algebraically closed field ...
1
vote
1answer
59 views

Analog of holomorphic Lefschetz fixed point theorem for smooth algebraic varieties

If $X$ is a compact complex manifold and $f: X \to X$ is a holomorphic map with isolated nondegenerate zeroes. Then there is a version of Lefschetz fixed point formula with traces on Dolbeaut ...
1
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0answers
35 views

Interpretation of $\Omega_{A/k} \simeq A \otimes_k I/I^2$ for affine group schemes

I'm learning some group scheme stuff and there's the following result: If $A$ is Hopf $k$- algebra, then $\Omega_{A/k} \simeq A \otimes_k I/I^2$, where $I$ is the augmentation ideal. I know the ...
1
vote
1answer
54 views

etale neighborhoods

I've read that quasi-compact etale morphisms of schemes over a not necessarily algebraiclly closed field $F$ (I'm happy to take $F$ a field of char $0$) are the algebraic analogs of local ...
1
vote
1answer
41 views

Finding the irreducible components.

Let $X$ be an algebraic set of the affine $3$-space $\mathbb{A}^{3}$ given by $x_{1}^{2}-x_{2}x_{3} = x_{1}x_{3}-x_{1}=0$. Find the irreducible components of $X$. I can easily figure out that we ...
1
vote
0answers
32 views

Rational equality modulo $p$

Assume that we have two rational expressions $f,g\in \mathbb{Z}(x,y_1,\ldots,y_n)$ with the property that the variable $x$ can only appear in the numerator of $f$ while only in the denominator of $g$. ...
0
votes
0answers
31 views

A question about locally free differential sheaf and regular local ring

Let $B$ be a local ring containing a field $k$ isomorphic to its residue field. Assume furthermore that $B$ is a localisation of a finitely generated $k$-algebra. Then $Ω_{B/k}$ is a free $B$-module ...
0
votes
2answers
68 views

Abelian category without enough injectives

What is an example of an abelian category that does not have enough injectives? An example must exist, but I haven't been able to find one. If possible, a brief explanation of why the abelian ...
3
votes
2answers
63 views

$\dim (D-P)=\dim (D)-1$

I'm trying to prove this question: Let $D$ be a divisor in $F|K$ such that $\dim (D)\gt 0$ and $0 \neq f\in \mathscr L(D)$. Thus $f\notin \mathscr L(D-P)$ for almost all $P$. Then show that ...
1
vote
1answer
53 views

A nonfree module which is locally free

The general context is trying to understand the Picard groups of various schemes, but this question focuses on affine schemes. Let $X=Spec A$ an affine scheme. What conditions does $A$ need to ...
1
vote
1answer
39 views

fibred product of groups of multiplicative type (or, more generally, of linear algebraic groups)

Let $M, M', M''$ be $k$-groups of multiplicative type, and let $M' \to M$ and $M'' \to M$ be morphisms of group schemes. Is the fibred product $M' \times_M M''$ a $k$-group of multiplicative ...
0
votes
0answers
28 views

Same number of generators and relations in a complete intersection, when?

I make this question a bit more general because i think as i put it, it will have no answer because there are too many maybe irrelevant details: Given $B$ an $A$-algebra, local, of finite type (that ...
1
vote
0answers
57 views

Local complete intersection ring

Suppose $R$ is a local Noetherian complete intersection ring that is a finite $A$-algebra, where $A$ is a DVR. If the module of differentials of $R$ is free as an $R/\mathfrak a$-module for some ...
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vote
0answers
36 views

a question on torsors

Let $\phi : Y \to X$ be an $X$-torsor under $G_X = G \times_k X$, where $Y,X$ are $k$-schemes and $G$ is an algebraic $k$-group. Let $U \subseteq X$ be an open subset, and let $W = \phi^{-1}(U)$. I'm ...
1
vote
1answer
100 views

Some questions about Hartshorne chapter 2 proposition 2.6

In Hartshorne chapter 2 proposition 2.6,Hartshorne shows that there is a fully faithful functor $t:\mathcal{Var}\rightarrow \mathcal{Sch}(k)$ from the category of varieties over $k$ to the category of ...