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. ...

learn more… | top users | synonyms (1)

3
votes
3answers
166 views

Problem in proving that $\mathbb{A}^2$ is not homeomorphic to $\mathbb{P}^2$

let $k$ be an algebraic closed field. All the spaces are equipped with the usual zariski topologies. All the proofs of this fact that I've seen rely on the fact that two lines in $\mathbb{P}^2$ ...
0
votes
0answers
35 views

Leray-Hirsch analogue of algebraic geometry

I want to use a Leray-Hirsch analogue of algebraic geometry to construct the chern classes. I am not sure how to prove the statement. Suppose that $E$ is a locally free sheaf of rank $r$ on $X$. I ...
1
vote
1answer
33 views

Representable open immersion of functors is a monomorphism

I have a question concering the proof of theorem 8.9 in Algebraic Geometry I (U. Görtz, T. Wedhorn). I will introduce what is needed. Let $S$ be a scheme, for an $S$-scheme $X$, we denote ...
2
votes
0answers
28 views

could someone help me with this particular detail in this article?

I'm reading this article and I was stuck in this part: I didn't understand his trick computing the orders at $P$. Remark: He defines $D=\inf\{div(\omega_{g-1}),div(\omega_g)\}$, i.e., ...
3
votes
1answer
77 views

Understanding the stack $B\mathbb{Z}$

Here, let $\mathbb{Z}$ be the group scheme whose functor of points is the constant functor which takes a connected affine scheme to the group $\mathbb{Z}$. I'm having a bit of trouble understanding ...
8
votes
2answers
231 views

Is the ring of p-adic integers of finite type over the ring of integers?

Denote by $\mathbb{Z}_p$ the ring of $p$-adic integers. Is $\mathrm{Spec}(\mathbb{Z}_p)$ of finite type over $\mathrm{Spec}(\mathbb{Z})$?
7
votes
3answers
167 views

What does the Tate module of an elliptic curve tell us?

I started studying elliptic curves, and I see that it is rather common to take the Tate module of an elliptic curve (or, of the Jacobian of a higher genus curve). I'm having a hard time isolating the ...
2
votes
1answer
35 views

Identifying the orbit space of the unitary group $U(n)$ in the compact symplectic group $Sp(n)$

Let $Sp(n)$ be the compact symplectic group. Let $U(n)$ the unitary group, and $O(n)$ the orthogonal group. What is $Sp(n)/U(n)$? What is $U(n)/O(n)$? I obtain that ...
3
votes
1answer
41 views

branched covering factors through a primitive one

I'm struggling with an assertion I found in an article I'm reading. A projective complex curve $X$ is rationally uniformized by radicals if there exists a branched covering $X\to \mathbb{P}^1$ such ...
3
votes
0answers
219 views

There is some intuitive idea of Pascal's 's theorem in Projective Geometry?

In projective geometry, Pascal's theorem (formulated by Blaise Pascal when he was 16 years old) determines that a hexagon inscribed in a conic, the lines that contain the opposite sides intersect in ...
0
votes
0answers
26 views

How to check whether a linear map on integral domains is a formal derivative

I have an elementary question on formal derivatives. Assume $A=K[X,Y,Z]/I$ is an integral domain (for example $I$ is a prime ideal and K is the field of rationals). Let $d:A\to A$ be a linear map. Is ...
1
vote
1answer
27 views

connection between discrete valuation rings and points of a curve.

Let $C$ be a projective irreducible non-singular curve over a field $k$ and let K be its function field. It applies that $(k[X,Y]/I(C))_{(X-a,Y-b)}$ (i.e. the localization of $k[X,Y]/I(C)$ at ...
2
votes
2answers
39 views

If $f\in k(\mathbb{A}^1)$ and $f^2\in k[\mathbb{A}^1]$, then is $f\in k[\mathbb{A}^1]$?

Suppose $k$ is algebraically closed, and $f\in k(\mathbb{A}^1)$ is in the field of rational functions over the variety $\mathbb{A}^1$. If we also know that $f^2$ is in the coordinate ring ...
1
vote
2answers
48 views

examples of interpreting schemes (Eisenbud)

I am having trouble understanding the role primary decomposition plays in ``interpreting'' the geometric picture of a scheme. Here are the examples I am struggling with from Eisenbud's Commutative ...
2
votes
1answer
17 views

connected linear algebraic group over the algebraic closure of a field

Let $G$ be a connected linear algebraic group over a field $k$ of characteristic 0. A paper I'm reading seems to imply that $\overline{G}:= G \times_k \overline{k}$ will also be connected, but I don't ...
2
votes
1answer
272 views

Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space ...
4
votes
1answer
26 views

coefficients of the zeta function of curve over a finite field $\mathbb{F}_q$

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
2
votes
0answers
22 views

another representation of the zeta function of a curve over a finite field

Let $C$ be a non-singular curve over $\mathbb{F}_q$. Denote by $d$ the degree map from the group of divisors to $\mathbb{Z}$ and denote by $P$ the set of prime divisors w.r.t. to the function field. ...
0
votes
1answer
18 views

Is the closed subgroup of any semisimple linear algebraic group semisimple?

Let $S$ be a semisimple linear algebraic group $/K$, with $K$ a field and $char K = 0$. Let $H \leq S$ be a closed subgroup $/K$. Is $H$ semisimple?
3
votes
1answer
80 views

What is GAGA for dimension 1 ? (Historical Question)

I know Riemann surfaces are actually algebraic curves, i.e. all Riemann surfaces can be simply embedded into some projective space $\mathbb{P}^n$. But this doesn't indicate me more correspondences ...
1
vote
0answers
33 views

Chow lemma + resolution of singularities

I am trying to understand the following simple reduction in Deligne's Theorie de Hodge II. He works in the category of schemes of finite type over $\mathrm{Spec}(\mathbf{C})$. Let $f : X \to S$ be a ...
0
votes
0answers
29 views

What is the kernel of $I/I^2 \to \Omega_{\mathbb P^{n}/k} \otimes \mathcal O_X$?

Recall that if $X \subset \mathbb P^n$ is a smooth projective variety, we have the conormal sequence of locally free sheaves on $X$ (here $I$ is the ideal sheaf of $X$): $$ I/I^2 \xrightarrow{\delta} ...
1
vote
1answer
19 views

Subgroup of an affine, algebraic irrducible group.

Let $G\subseteq GL_n(\mathbb{C})$ be a irreducible, affine, algebraic group (Zariski-Closed). Moreover let $H \subseteq G$ be a finite normal group. I want to show that $H \subseteq Z(G):=\{g \in ...
2
votes
1answer
45 views

Smallest Convex Hulls ($n$-Simplex) of $n+2$ Points in $\mathbb{R}^n$

"Given $n$, find the minimal value of $k$ with the following property: Any $k$ (distinct) points in $\mathbb{R}^n$ can be partitioned into two disjoint subsets so that the intersection of the convex ...
1
vote
0answers
29 views

A question about Hilbert's Nullstellensatz

We know that Hilbert's Nullstellensatz is valid for $\Bbb{C}[X]$, as $\Bbb{C}$ is a closed field. Let us consider the ideal $(x+y,x-y)\subset \Bbb{C}[x,y]$. Clearly, $Z((x+y,x-y))=\{(0,0)\}$. Now ...
0
votes
1answer
23 views

Condition for dim of the Euclidean space with orthogonal basis

I would like to show that if the orthogonal basis of the $\Bbb R^n$ Euclidean space with the standard dot product has the vectors whose elements are exclusively $1$ or $-1$, then $n \le 2$ or $n$ is ...
2
votes
1answer
37 views

When is an holomorphy ring a PID?

I posted this question on mathstackexchange but I realized it is probably more suitable for mathoverflow. I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. ...
0
votes
0answers
30 views

quadric in $P^{3}$

I have a difficulties to show the followings: Let k be an algebraically closed field. A quadric in $P^{3}(k)$ is a projective algebraic set of the form Q = V(F), where F is an irreducible polynomial ...
2
votes
1answer
43 views

Is the Projectivization of a coherent sheaf on a reduced Noetherian scheme reduced again?

Let $X$ be a reduced Noetherian scheme and $\mathcal{F}$ a coherent sheaf on $X$. What i am wondering is: Is it true or not that the projectivization ...
8
votes
1answer
318 views

Is an ideal generated by multilinear polynomials of different degrees always radical?

Definition. A polynomial $f\in\Bbbk[x_0,\ldots,x_n]$ is called multilinear if $\deg_{x_i}(f)=1$ for each $0\le i \le n$. In other words, $f$ is linear in each variable. If $f$ is homogeneous of degree ...
0
votes
0answers
28 views

Some basic questions about fibered surfaces

I get stuck at the section 8.3 Fibered Surfaces of Qing Liu's book Liu: Algebraic Geometry and Arithmetic Curves and I feel strange that it is not easy to find many other books or papers discussing ...
6
votes
0answers
78 views

Duals of representations of affine group schemes, in particular $\mathrm{GL}_n$

Duals of representations of affine group schemes Let $R$ be a commutative ring. If $G$ is a group and $V$ is a dualizable i.e. finitely generated projective $R$-module on which $G$ acts, then it is ...
3
votes
1answer
110 views

Definition of a Elliptic curve

I've seen two different definitions of an elliptic curve. 1) The first one being that it is a nonsingular projective curve of genus 1. 2) The other definition nonsingular projective curve of ...
3
votes
1answer
107 views

How to prove that $CP^4$ cannot be immersed in $R^{11}$

Please let me know how to prove that $CP^4$ cannot be immersed in $R^{11}$. I know a proof using an integrality theorem for differentiable manifolds but I want to know if a more direct and simple ...
1
vote
1answer
48 views

Integral extensions with finitely generated k-algebras

I have $k$ a field, and I am assuming that the finitely generated $k$-algebra $K = k[x_1,x_2]$ is also a field. I am trying to prove Zariski's lemma in this case, by seeing first that $K$ is an ...
1
vote
1answer
28 views

Restriction of scalars of a torus

Let $k$ be a number field, $l/k$ be a finite extension, and $T_{/l}$ be a linear algebraic torus over $l$. Is $R_{l/k}(T_{/l})$ a linear algebraic torus over $k$? Here $R_{l/k}$ is the restriction ...
6
votes
2answers
63 views

Is the set of complex solutions to $x^2+y^2 = 1$ isomorphic to $\mathbb{C}^*$?

In his article about Grothendieck, Edward Frenkel states that the set of complex solutions to the equation $x^2+y^2 = 1$ is "a plane with one point removed." I'm curious how this can be made precise. ...
2
votes
1answer
31 views

Coextension of Scalars vs Direct and Inverse Images

Given a morphism of schemes $f:X\to Y$, we get a canonical direct image morphism of modules, $f_*$, with $f_*\mathscr{G}(U)=\mathscr{G}(f^{-1}(U))$. This is an $\mathscr{O}_Y$-module via the pullback ...
3
votes
2answers
161 views

General quadratic diophantine equation.

Here is my problem: I am given a general quadratic diophantine equation: $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$ where $x$ and $y$ are variables with integers $a,b,c,d,e,f$. I have to show that if the ...
1
vote
2answers
29 views

An example of an ideal needed.

Let $A\subseteq \Bbb{C}[X]$ be an ideal. I am looking for an example of $A$ such that $A\subsetneq I(Z(A))$. Here $Z(A)$ is the set of zeroes that are satisfied by all polynomials in $A$ and ...
1
vote
1answer
61 views

Is quotient under $S_4$ action on “cube” representation a flat morphism?

Consider a three-dimensional irreducible representation $V$ of $S_4$, corresponding to symmetries of cube. Let $p$ be canonical projection $p: V \rightarrow V/S_4$. My question: is $p$ flat? I want ...
1
vote
1answer
37 views

Global sections of canonical sheaf and genus

Let $k$ be algebraically closed and C a curve. Then $g(C) = h^0(C, \omega_C)$. I want to show that if all the degree $0$ line bundles are trivial, then $g = 0$. I can sort of see that if I think of ...
0
votes
0answers
29 views

Clarification of exercise 7.3F in Vakil's exercise FOAG

I come across this question which I have trouble understanding. It says (7.3F) Suppose $Z$ is a closed subset of an affine scheme $Spec A$ locally cut out by one equation, (In other words, $Spec ...
1
vote
1answer
30 views

Algebraic set in $\Bbb R^2$

Why $C=\{(x,\sin x): x\in\mathbb R\}\subset\mathbb R^2$ not is algebraic set? i.e. $C$ are the zeros of $S\subset \mathbb R[X,Y]$. Thank you, for any suggestion.
2
votes
3answers
118 views

The algebraic set is irreducible

If $V$ is an algebraic set of $K^n$, I want to show that $V$ is irreducible exactly when $I(V)$ is a prime ideal. That's what I have tried: We suppose that $V$ is not irreducible. Then, it can be ...
0
votes
0answers
40 views

Definition of $\sum_{a \in A} I_a$

If $(I_a)_{a \in A}$ a family of ideal of $K[x_1,x_2, \dots, x_n]$, I have the following definition in my notes: $$\sum_{a \in A} I_a=\{ a_{i1}+a_{i2}+ \dots+ a_{ij} | a_{ij} \in I_{a_j} \}$$ Is ...
14
votes
2answers
593 views

History of Algebraic Geometry: Motivation behind definition of schemes

I am trying to read an article by Jean Dieudonne which talks about development of Algebraic Geometry. The article was being published in the journal "Advances in Mathematics" Volume 3, Issue 3, Pages ...
1
vote
1answer
36 views

Why does the Hecke correspondence preserve principal divisors?

Let $p$ be prime not dividing $N$. Consider the Hecke correspondence $T_p$ inducing a set valued function on $X_0(N)$. I'd like to understand why it acts on $\text{Pic}^0$, and so I'd like to know why ...
1
vote
1answer
33 views

Why a semi-stable non stable bundle $E$ is S-equivalent to $L_1\oplus L_2$

Let $M(2,d)$ be the set of all vector bundles of rank $2$ and degree $d$ over a smooth projecitve curve of genus $g\geq 3$. Let $M(2,0)^s$ and $M(2,0)^{ss}$ be the stable and semistable vector ...
1
vote
1answer
52 views

Is a flat coherent sheaf over a connected noetherian scheme already a vector bundle?

Let $A$ be a connected noetherian ring (not necessarily irreducible), $M$ be a finitely presented flat $A$-module. Then $M_{\mathfrak{p}}$ is a free $A_{\mathfrak{p}}$-module for each $\mathfrak{p} ...