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)

0
votes
1answer
21 views

Surjectivity of ring homomophism induced by Frobenius endomorphism

Denote by $F_q$ the finite field with $q$ elements, and denote by $\bar{F_q}$ its algebraic closure. Let $V$ be an affine $\bar{F_q}$-variety and $F$ be the Frobenius endomorphism corresponding to an ...
3
votes
1answer
62 views

Nilpotents after tensoring with a field

Let $A \to B$ be a homomorphism of commutative rings with unit. Let $A_{\text{red}}=A/ \sqrt{(0)}$ and $B_{\text{red}}=B/ \sqrt{(0)}$ be the corresponding reduced rings. Now let $A_{\text{red}} \to K$ ...
1
vote
0answers
22 views

Class preserving Autpmorphism

This time I am having interest in theory of Class preserving Automorphism and central preserving Automorphism and some topics related to Automorphism of groups. So Could you please tell me some ...
0
votes
1answer
30 views

Veronese image not contained in proper linear subspace

I'm studying the Veronese Map from Shafarevich Book. And I want to prove the following problem (Ex 10 Section 4.4 Shafarevich Basic Algebraic Geometry) Prove that the Veronese image ...
3
votes
1answer
57 views

Exercise 1.2 in Hartshorne Chapter I

I got most of this exercise but there are a few things I am confused about: Let $A = \{ (t,t^2,t^3) : t \in k\}$. Show that $A$ is closed and irreducible of dimension one, find generators of the ...
1
vote
2answers
42 views

Is every fiber of a morphism between varieties of pure dimension?

Suppose $f\colon X\to Y$ a morphism of varieties with connected fibers, is it true that all the fibers have pure dimension?
1
vote
3answers
39 views

Linear section of a smooth variety

Let $X \subset \mathbb{P}^{N}$ be a non-degenerate smooth variety with positive dimension. Take $x_{1}, \ldots, x_{n}$ general points on $X$, with $\mathrm{codim}(X) \geq n - 1$. Denote by $P$ the $(n ...
8
votes
1answer
59 views

An element of $f$ of a function field such that $P$ is the only pole of $f$.

Let $F$ be a function field in one variable over a field $k$. Let $S$ be a nonempty finite subset of all places of $F$. Prove that if $P \in S$, there is an element $f$ of $F$ such that $P$ is the ...
1
vote
1answer
56 views

Smoothness for morphism of schemes

Let $X \to Y$ be a projective morphism of schemes of finite type with $Y = Spec(R)$, where $R$ is a dvr. For this morphism to be smooth, is it sufficient to check smoothness on only closed points of ...
0
votes
1answer
30 views

How can we show the torsion subgroup of a group is pure?

I found a definition of pure subgroup: Let $G$ be an abelian group and $H\leq G$. $H$ is a pure subgoup of $G$ if $\forall h \in H$, if $h$ is divisible by $n$ in $G$, then it is divisible by $n$ in ...
0
votes
0answers
26 views

Classification of local and semi-local rings in function fields

Let $C$ be a non-singular algebraic curve over an algebraically closed field $k$, and $F$ a function field of this curve. It is well-known that non-trivial discrete valuation rings of $F$ correspond ...
1
vote
0answers
38 views

How are isomorphisms shown on open sets or using category theory in algebraic geometry? [closed]

I may have seen a few examples of how isomorphisms are shown in algebraic geometry using open sets such as $D\left(f\right)$ and/or category theory methods, but I don't have understanding of the ...
0
votes
1answer
32 views

Dimension is an invariant of isomorphism class of projective varieties

I'm trying to understand the solution to the following problem - showing that the dimension of a projective variety is an invariant of it's isomorphism class. I'm struggling a bit though. My idea ...
1
vote
0answers
26 views

Why is this scheme $Y$ an affine bundle?

Suppose you have $F$ a subbundle of a vector bundle $E\to X$ over a scheme $X$. Recall that there is a scheme $\varphi\colon Y\to X$ with the points of $Y$ over the point $x\colon\mathrm{spec}(A)\to ...
1
vote
1answer
31 views

Existence of induced map on Divisor Class Group?

Let $f: X \rightarrow Y$ be a morphism of noetherian, integral schemes, regular in codimension 1 (so we can talk about Weil divisors). I am wondering whether there is an induced map on divisor class ...
0
votes
0answers
61 views

Singularities in the weighted projective space

Is there an explicit criterion for checking that a hypersurface $f=0$ of degree $d$ and in $\mathbb{P}(a_0,\ldots,a_n)$ is smooth ? I could not convince myself that the criterion $\nabla f\neq 0$ ...
1
vote
1answer
31 views

Characterisation of “projective $k$-algebras”

For my thesis, I'm defining affine $k$-algebras to be reduced, finitely generated $k$-algebras--each of which turns out to be isomorphic to the quotient of a polynomial ring by a radical ideal. I'm ...
2
votes
0answers
32 views

Pullback of box products of invertible sheaves

Let $S, T$ be graded $A$-algebras (where $S_0 = A = T_0$) finitely generated in degree $1$ and let $M$ be a quasi-finitely generated graded $S$-module, $N$ a quasi-finitely generated graded ...
1
vote
2answers
37 views

affine scheme X with exactly two points and dimension 1

I am looking for an affine scheme X with exactly two points and dim $X=1$. I am not sure if it exists, but it seems to me that it does. I already (hopefully being right) that such a scheme with ...
1
vote
1answer
27 views

Finding angle of rotation of an ellipse

Suppose I have the ellipse $$ x^2 -2xy +4y^2 = 1 $$ How can I find the angle at which this ellipse is rotated? I have tried to assign $x=\cos\theta, y=0.5\sin\theta$ but I don't know if that's the ...
1
vote
1answer
51 views

27 lines on Fermat surface

I want to describe the $27$ lines on the Fermat surface and have found the information below. I don't understand the last part. How is it possible to go from $9$ different lines to $27$ different ...
4
votes
0answers
58 views

Geometry of cubic 3-fold

I'm having some questions about the geometry of the cubic 3-fold. Every variety is over $\mathbb C$. Take $Y$ a smooth cubic 3-fold in $\mathbb P^4$ and $E$ a curve of degree 6 and genus 1 contained ...
3
votes
1answer
31 views

Test for a $G$-torsor to be trivial?

I just have a very short question, why is a $G$-torsor trivial precisely when it has a section?
4
votes
2answers
98 views

What are the closed points of $\mathbb{A}_{\mathbb{R}}^2 = \operatorname{Spec}(\mathbb{R}[x,y])$?

I am trying to find all the closed points of $\mathbb{A}_{\mathbb{R}}^2$. After a quick google research, I found that $\mathbb{A}_{\mathbb{R}}^2 = \operatorname{Spec}(\mathbb{R}[x,y])$ and then all ...
2
votes
2answers
54 views

How can I prove this: “$\mathbb{P}^{n} \times \mathbb{A}^{m}$ is not affine variety$”

I'm trying prove this statement, but I can't realize what I need to show first. First I wanna take some isomorphism using one thing I know: "$\mathbb{A}² \backslash \{0,0\}$ is not affine. Someone ...
2
votes
1answer
26 views

Definition of a $k$-structure

I came across the following definition: Let $\Omega$ be algebraically closed, $k \subseteq \Omega$ a subfield, and $V$ an $\Omega$-vector space. A $k$-structure on $V$ is a $k$-vector space $V_k$ ...
0
votes
1answer
17 views

Irreducible decomposition of varieties vs primary decomposition of ideals

I'm new to working with varieties, and the statement mentioned below is left as an exercise, but I'm having some difficulty trying to prove it. Let $R=K[x_1,...,x_n]$. If $X=X_1\cup ... \cup X_n$, ...
1
vote
1answer
32 views

Show that the variety $V(I(X))=X$

In the ring $R=K[x_1,...,x_n]$, the variety of an ideal is defined as $V(I)=\{(a_1,...,a_n)\in K^n|f(a_1,...,a_n)=0, \space\forall f\in I\}$ The ideal of a variety is defined as $I(V)=\{f\in ...
0
votes
1answer
20 views

Weighted projective space

Here is one example from Cox's lecture notes: I really don't know how to use $M$ to define an automorphism. Intuitively, I can rescale the first coordinate to $1$ and I think the isomorphism is ...
0
votes
1answer
60 views

Theorem** on page 288 from Geometry of algebraic curves by Arbarello, Cornalba, Griffiths, Harris

1Let $V$ a $2n$ complex vector space and take on $V$ a quadratic form. Now define $$ \Sigma=\{\Lambda:Q(\Lambda,\Lambda)\equiv0 \} \subset Gr(n,2n)$$ where $\Lambda$ is a maximal subspace i.e it is ...
1
vote
4answers
112 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
1
vote
1answer
47 views

Does every algebraic variety admit a local parametrization at every non-singular point?

I am reading a text in which the first sentence of the proof of a theorem is: Let $X(t)=X(t_{1},\ldots,t_{n})$ be a local parametrization of the algebraic variety $X$... I guess that every ...
0
votes
0answers
30 views

The complement of the image of the zero section is still a $\mathbb{G}_m$-torsor?

This came up while doing some reading Schneps text on Galois Groups and Fundamental groups, but it's glossed over. In any case, suppose that you have a line bundle over a scheme $L\to X$, with zero ...
1
vote
1answer
25 views

Hartshorne's Example IV.3.3.5

In Hartshorne's book Algebraic Geometry he says in Example 3.3.5 in Chapter IV (p. 309): If $X$ is a plane curve of degree 4, then $D=X.H$ is a very ample divisor of degree 4. Here $H$ is a ...
1
vote
0answers
30 views

tangent cone to the set

I'm supposed to solve this problem: Let us consider the set $M=\{(x, \sin{x}):x\in\mathbb{R}\}\cup\{\big(\cos(x)-1,x\big):x\in\mathbb{R}\}$ The question is to find the tangent cone to the set $M$ in ...
1
vote
0answers
32 views

Divisors corresponding to hypersurfaces in Projective space

I'm looking at hypersurfaces on $\mathbb{C}\mathbb{P}^2$. That is, the zero set of an irreducible homogeneous polynomial $f(x_0, x_1, x_2) = 0$. This corresponds to a divisor, $D_f$, let's say which ...
7
votes
0answers
87 views

$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup ...
1
vote
2answers
42 views

Affine open sets of projective space and equations for lines

I am reading Introduction to Algebraic Geometry by Smith et al. and I have some questions about some vocabulary that they use but that is not explicitly defined (I guess it is probably obvious and I ...
1
vote
1answer
35 views

Interpretation of a short exact sequence from elliptic curves in terms of torsors

Consider some elliptic curve $E$ over a number field $k$. Then for any prime $p$ there is a short exact sequence $$ 0 \to E(k)/pE(k) \to H^1(k,E[p]) \to H^1(k,E)[p] \to 0. $$ Now, $H^1$ has an ...
1
vote
0answers
24 views

deducing irreducibility from intersections with hyperplane complements

Let $X$ be a projective variety of $\mathbb{P}^n$. Suppose that for any hyperplane complement $U$ of $\mathbb{P}^n$, $X \cap U$ is irreducible. Then i want to prove that $X$ must be irreducible. Here ...
1
vote
0answers
25 views

Inclusionwise maximal linear subvarieties of a projective variety

Let $X\subseteq\mathbb P^n$ be a complex, projective variety. A linear subspace $L\subseteq\mathbb P^n$ will be called a maximal linear subspace of $X$ if $L\subseteq X$ and for any linear subspace ...
3
votes
2answers
59 views

Deforming line bundles on abelian varieties

Let $X$ ba an abelian variety over $\mathbb C$. I would like to understand how line bundles on $X$ deform. The obstructions to deform line bundles lie in $$\textrm{Ext}^2(L,L)=H^2(X,\mathscr O_X).$$ ...
3
votes
1answer
134 views

The assignment $R\mapsto\operatorname{Iso}_{R\text{-alg}}(A\otimes_k R,M_n(R))$ is a scheme?

Let $A$ be a central simple algebra over some field $k$, with degree $n$. There is a functor $F$ defined by the assignment, for a commutative ring $R$, $$ ...
5
votes
0answers
53 views

Some questions about reduction of elliptic curves

Let $E \rightarrow S$ be an elliptic curve (i.e, a smooth proper curve of genus 1). If $S = \text{Spec (K)}$ where $K$ is a local field, the usual way of doing a reduction at a prime $\mathfrak{p} = ...
3
votes
0answers
48 views

Solving exercise 1.10 in Silverman's AEC

Please note that although there is a very similarly titled question Exercise 1.10 from Silverman "The Arithmetic of Elliptic Curves" this question received no answers. Let $p$ be an odd prime and ...
1
vote
1answer
36 views

Dimension of linear system of divisor of two points on curve of genus greater than 2

This should not be hard, but I am stuck on it nonetheless, so I would much appreciate a solution. Suppose $C$ is a projective non-singular curve of genus $g\geq 2$ and $P,Q$ are distinct points on ...
4
votes
1answer
46 views

Separated scheme stable under base extension.

Given a separated scheme morphism $X\to Y$, and a morphism $Z \to Y$, Hartshorne proves that the extension $X\times_YZ \to Z$ is also separated, as long as the schemes involved are Noetherian. The ...
1
vote
1answer
30 views

Holomorphic maps between smooth algebraic curves

I am looking for a reference for the following statement: Let $X$ be a smooth projective curve over $\mathbb{C}$. Every holomorphic function $f: X \to \mathbb{P}^1_{\mathbb{C}}$ is in fact a morphism ...
0
votes
0answers
36 views

Affine morphism, Harsthorne ex 5.17 a)

$\newcommand{\Sp}{\text{Spec}}$ Hello, I'm stuck with one question in Hartshorne, exercise 5.17. A morphism between schemes $f: X \to Y$ is affine is there is an open cover $Y = \bigcup V_i$ such ...
0
votes
1answer
25 views

Picard group of Segre Embedding of $\mathbb{P}^1 \times \mathbb{P}^1$

Let $X = V(xy-zw) \subset \mathbb{P}^3$ (the variables are $x,y,z,w$). I know from various sources that $Pic(X) \cong \mathbb{Z} \oplus \mathbb{Z}$, where generators are the two lines $l_x = V(x,w) ...