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
2answers
459 views

Why is the disjoint union of a finite number of affine schemes an affine scheme?

We know that the disjoint union of an infinite number of affine schemes is not an affine scheme since the underlying topological space is not quasi-compact. But how do you show that the disjoint ...
3
votes
1answer
336 views

Chapter V of Grothendieck's EGA

Grothedieck wrote, in the introduction of his EGA, Chapter V would be Procedes elementaires de construction de schemas(Elementary procedures for construction of schemes). I wonder what he meant by it. ...
3
votes
4answers
380 views

Local ring at a non-singular point of a plane algebraic curve

Let $k$ ba field. Let $F(X, Y)$ be a non-constant polynomial in $k[X, Y]$. Suppose $F(0, 0) = 0$. Then $F(X, Y)$ is of the form $aX + bY +$ higher degree terms. Suppose $aX + bY \neq 0$. Let $A = k[X, ...
3
votes
1answer
285 views

Left adjoint and right adjoint/ Nakayama isomorphism

I am reading a paper "2-vector spaces and groupoid" by Jeffrey Morton and I need a help to understand the following. Let $X$ and $Y$ be finite groupoids. Let $[X, \mathbb{Vect}]$ be a functor ...
3
votes
1answer
108 views

How can function fields have different degrees over the projective line

I'm confused. Let $X$ be a curve over a field $k$. Let $K= K(X)$ be its function field. Then, $K(X)$ is a field. Each non-constant morphism $f:X\to \mathbf{P}^1_k$ gives a field extension ...
3
votes
1answer
203 views

A Problem in Shafarevich's book

I have encountered a problem in Shafarevich's book and I have no clue: Let $X$ be a hypersurface given by the equation $f_{m-1}(x_1,\cdots, x_n) + f_m(x_1,\cdots, x_n) = 0$ where $f_{m-1}$ and $f_m$ ...
3
votes
1answer
95 views

find $\dim X\times \mathbb{P}^{2}$

I tried to solve this question but not sure that my solution was the right one. Help me please with this question: Find $\dim X\times \mathbb{P}^{2}$, where $X=\left \{ ...
3
votes
1answer
109 views

How to handle group schemes by points?

I find it is very inconvenient to handle group schemes by its defination(i.e. everything is defined by morphism). And I have noticed that for group varieties, one can treat them as actual groups(i.e. ...
3
votes
1answer
262 views

Hypersurface becomes an hyperplane after embedding

Let $X$ be an hypersurface of degree $k$ in $\mathbb{P}^{n}$, why the equation defining $X$ becomes linear in the Veronese coordinates? More precisely I want to understand the last paragraph of the ...
3
votes
1answer
103 views

For curves, is being defined over a number field invariant under birational equivalence

Suppose that a (connected) Riemann surface $X$ is birational to a Riemann surface $Y$ which can be defined (algebraically) over the field of algebraic numbers. Does this imply that $X$ itself can be ...
3
votes
2answers
169 views

Does there exist a finite morphism of algebraic curves such that…

Let $K\subset L$ be a finite field extension. Let $X$ and $Y$ be (smooth projective geometrically connected) curves over $L$. Let $f:X\to Y$ be a finite morphism of curves over $L$. Assume that ...
3
votes
2answers
622 views

Finitely-generated $k$-algebras and their relationship with affine coordinate rings

Let $k$ be an algebraically closed field, $A = k[x_1, ... , x_n]$. For $Y \subseteq \mathbb A^n$, define $I(Y) = \{f \in A| f(P) = 0 \ \forall P \in Y\}$ Hartshorne's Algebraic Geometry, p. 4-5, says ...
3
votes
1answer
137 views

Question concerning the Hodge conjecture.

Let $X$ be a projective complex manifold of (complex) dimension $n$. Let $A \subset X$ be a closed submanifold and $[A]$ be the Poincare dual to its fundamental class. Can you please answer the ...
3
votes
1answer
119 views

The set of arithmetical numbers

Define $x\in\mathbb{R}$ to be arithmetical number if the set $\{\langle p, q \rangle \in \mathbb{Z}^2 : \frac{p}{q} < x\}$ is an arithmetical set. Define $x\in\mathbb{C}$ to be arithmetical number ...
3
votes
1answer
172 views

Compact Sets in Projective Space

Consider the projective space ${\mathbb P}^{n}_{k}$ with field $k$. We can naturally give this the Zariski topology. Question: What are the (proper) compact sets in this space? Motivation: I ...
3
votes
3answers
185 views

Quotient of a variety and orbits

Suppose a group G acts on a variety X and a quotient exists, that is, we have a variety Y and a regular map $\pi : X \rightarrow Y$ so that any regular map $\varphi :X \rightarrow Z$ to another ...
3
votes
3answers
274 views

A Noetherian Ring with Discrete Spectrum is Artinian

I'm trying to solve an exercise. I should prove that if $R$ is a notherian ring and $\operatorname{Spec}(R)$ is discrete then $R$ is artinian. I think it is enough to show that $\dim R=0$ ...
3
votes
2answers
292 views

What is the tensor product of a sheaf and a module?

The following object was studied in section III.12 of Hartshorne's Algebraic Geometry book. Let $A$ be a noetherian ring, $Y=\mathrm{Spec}A$, and $M$ be an $A$-module. Let $f:X\to Y$ be a morphism, ...
3
votes
1answer
173 views

Tangent space of quasi-projective varieties

If $X$ is a quasi-projective variety and $X_i,\;\;i=1,\ldots,k\;$ are its irreducible components, then why $$\mathrm{dim}\;T_{X,x}=\mathrm{max}_{i=1,\ldots,k}\;(\mathrm{dim}\;T_{X_i,x})?\qquad \qquad ...
3
votes
1answer
140 views

Irreducible subspace of $\mathbb{A}^2$

Let $X:=V(x^m-y^n)$ be a subspace of $\mathbb{A}^2$. How can I prove that if $(n,m)=1$ then $X$ is irreducible? I think that it is isomorphic to $\mathbb{P}^1$ but I can't prove that.
3
votes
2answers
125 views

Showing a (relatively simple) set of polynomial zeros in projective space is irreducible

I'm teaching myself a little algebraic geometry and I was hoping you could help me with an exercise. I have my head around affine spaces alright but I am having a little more trouble with projective ...
3
votes
2answers
177 views

find birational maps of a hypersurface onto $\mathbb{A}^2$

Consider $$Y = V(y^2-x^3) \subseteq \mathbb{A}^2$$ Now, $\phi: \mathbb{A} \to Y, t \mapsto (t^2,t^3)$ is a birational map, but the pullback $\phi^\ast: K[x,y]/(y^2-x^3) \to K[t], x \mapsto t^2, y ...
3
votes
1answer
235 views

every divisor of degree $0$ on a smooth cubic curve $\mathcal{C}\subseteq\mathbb{P}^2$ is equivalent to $A-A_0$ for a fixed $A_0\in\!\mathcal{C}$

NOTATION: Let $\mathcal{C}=\mathcal{V}(F)\subseteq\mathbb{P}^2$ be a curve of degree $3\!=\!deg(F)$ with no singularities and let $A_0\!\!\in\!\mathcal{C}$ be fixed. Let $Div(\mathcal{C})$ denote the ...
3
votes
1answer
90 views

Is the product of two open embeddings of schemes an open embedding?

Question as in title. I only really need the special case where one of the open embeddings is the identity, but the more general case would be useful to know. Edit - By product I mean: given $U\to X$ ...
3
votes
1answer
2k views

How does a hyperplane become a linear bundle?

As we know,a hyperplane can seem as a divisor,and a divisor can become a linear bundle,I want to know what the structure of linear bundle is. For example, the hyperplane is given by $a_0 z_0+a_1 ...
3
votes
2answers
193 views

A polynomial of degree $k$ vanishing at $kd+1$ points on a rational normal curve in $\mathbb{P}^d$ must vanish on the whole curve

This is asserted in Exercise 1.15 of Joe Harris's algebraic geometry book (Algebraic Geometry: A First Course, Pg. 11 in my copy). This result struck my fancy but I'm unable to solve it myself or find ...
3
votes
1answer
269 views

What is a G-Galois Branched Cover

What is, in the language of Schemes, a G-galois branched cover?
3
votes
2answers
119 views

The group $E(\mathbb{F}_p)$ has exactly $p+1$ elements

Let $E/\mathbb{F}_p$ the elliptic curve $y^2=x^3+Ax$. We suppose that $p \geq 7$ and $p \equiv 3 \pmod {4}$. I want to show that the group $E(\mathbb{F}_p)$ has exactly $p+1$ elements. I was ...
3
votes
1answer
51 views

“Twist” of $\mathbb P^n_K$ through a field automorphism.

This question is closely related to this recent one. Suppose that $s:X\longrightarrow\text{Spec}\, K$ is a variety over $K$ (i.e. a $K$ scheme, separated, proper and geometrically integral) and ...
3
votes
1answer
197 views

How could we show that the abelian group has $\text{ rank}=0$?

Let $E/\mathbb{Q}$ the elliptic curve $Y^2=X^3+p^2X$ with $p \equiv 5 \pmod 8$. Show that the abelian group $E(\mathbb{Q})$ has $\text{rank}=0$. Could you give me a hint how we could do this? It is ...
3
votes
1answer
149 views

Elliptic curve- Component of point

If $E/ \mathbb{Q}$ elliptic curve in the general Form of Weierstrass and $P=(x,y)$ a rational point of it, show that the first coordinate of the point $2P$ is $$ ...
3
votes
1answer
90 views

Riemann-Roch Theorem

Could somebody give a simple plain English explanation as to what the Riemann-Roch theorem is about to somebody who knows only standard one-variable complex analysis. Thanks.
3
votes
1answer
66 views

Is every smooth $\mathbb{R}$-variety isomorphic to an affine variety?

I sadly don't know anything about formal GAGA yet, but I am at least trying to follow my intuition as often as possible. In differential geometry we know that we can embedd every smooth ...
3
votes
2answers
87 views

Why is $\text{Mor}_{\mathrm{reg}}(*,W)\to \text{Hom}_{k-\mathrm{alg}}(k[W],k[*])$ not surjective when $W=\mathbb{A}^2\setminus\{(0,0)\}$.

Suppose we're working over an algebraically closed field $k$. If $V\subseteq\mathbb{A}^n$ and $W\subseteq\mathbb{A}^m$ are affine algebraic sets, there is a well known bijective correspondence $$ ...
3
votes
1answer
69 views

Is a coherent locally free sheaf isomorphic it's dual?

Hartshorne chapter II problem 5.1 a) is to prove that the double dual of a coherent locally free sheaf $\mathscr{E}$ over a ringed space $(X,O_X)$ is isomorphic to $\mathscr{E}$. This can be done by ...
3
votes
2answers
76 views

Is every $\mathcal{O}_X$-module homomorphism $\mathcal{O}_X^{\oplus n} \to \mathcal{O}_X^{\oplus m}$ given by a matrix?

For $A$ a ring, we know any linear map from $A^{\oplus n} \to A^{\oplus m}$ is given by a $m \times n$ matrix. For $(X,\mathcal{O}_X)$ a locally ringed space (or scheme), is every ...
3
votes
1answer
109 views

Geometrical description of maps of schemes

In preparation for an exam, I am trying to solve the following question: Describe geometrically all maps from $\operatorname{Spec}(\mathbb{C}[z]/(z^2))$ to $\operatorname{Spec}(\mathbb{C}[x,y])$. ...
3
votes
1answer
43 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 ...
3
votes
1answer
72 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 ...
3
votes
1answer
76 views

Integral Domain with exactly two Prime Ideals

I am not looking for someone to give me an explicit example. I want to work this out myself if possible. Trying to learn schemes by reading The Geometry of Schemes by Eisenbud and Harris. Problem I-5 ...
3
votes
1answer
125 views

A Question from Algebraic Geometry

For any two disjoint closed subsets $Y_1$ and $Y_2$ of $ \mathbb A ^n$ show that there exists $g \in\mathbb C [x_1, x_2, ..., x_n]$ such that $g(Y_1)=0$ and $g(Y_2)=1$.
3
votes
2answers
105 views

Computing irreducible components of algebraic set

Consider the algebraic set $V(X^2-YZ,X-XZ)$. Find the irreducible components of this set and show that $I(V)=(X^2-YZ,X-XZ)$. I reasoned that $X-XZ=0$ iff $X=0$ or $Z=1$. If $X=0$, we get $Y=0$ or ...
3
votes
1answer
83 views

Derived functor vs. spectral sequence

I heard many times that because of introducing derived category, we can avoid cumbersome spectral sequence. However, I don't quite understand its meaning. Here is a precise example people talking ...
3
votes
1answer
78 views

An example of ample sheaf with no global section

In viewing the tags about ample bundle with no global sections I found an example below: If $C$ is a curve of genus $2$, and $p,q,r$ are general points on $C$, then the bundle ...
3
votes
1answer
206 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
3
votes
1answer
54 views

A criterion for quasiaffine varities

If $X$ is a prevariety and the open sets $X_f:=\{x:f(x)\neq 0\}$ form a basis for the topology of $X$ as $f$ varies over $\Gamma(X,\mathcal{O}_X)$, then why is $X$ quasi-affine? The definitions are ...
3
votes
1answer
94 views

Motivating examples of Spec(R) where R is not a finitely generated k-algebra

A scheme is a ringed space locally isomorphic to an affine scheme. An affine scheme is the spectrum of a commutative unital ring together with a structure sheaf. What are examples of "interesting" ...
3
votes
2answers
68 views

How to determine a set of polynomials is algebraically indepedent or not?

I have a set of polynomials $$ a a_1, bb_1, c, c_1, ab, da_1 + bd_1, fa_1+cf_1+d_1e, eb_1+ce_1, de-bf, f_1, e_1, d_1. $$ Is there some software which can determine that they are algebraically ...
3
votes
1answer
146 views

Normal bundle of a line in a blow-up

Let $x\in X=\mathbb{P}^3$. Consider the blow-up $\widetilde{X}\to X$ of $x$ in $X$. Let $l\ni x$ be a line in $X$ and $\widetilde{l}$ is its strict transform in $\widetilde{X}$. How to prove that the ...
3
votes
1answer
93 views

When divisor self-intersection number equal to 0?

Let $D$ be a divisor on smooth rational surface $X$ (over $\mathbb{C}$) such that the linear system $|nD|$ has dimension 1 (fore some $n$), has no fixed components and base points. How can I show ...