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)

10
votes
1answer
741 views

The Picard Group of the Affine line with double origin

Let $X$ be the affine line with double origin over a field $k$. It is the scheme obtained gluing two copies of the affine line $\mathbb{A}^1_k$ along the open sets $U_1 = U_2 =\mathbb{A}^1_k - (x)$, ...
9
votes
1answer
437 views

Learning projective geometry

My ultimate goal is to learn some algebraic geometry with the more concrete immediate goal of understanding things like how $\mathbb R\mathrm P^2$ is embedded in $\mathbb C\mathrm P^2$ or how $\mathbb ...
8
votes
2answers
253 views

When characteristic classes determine a bundle?

Let $k$ be a natural number and $F$ be $\mathbb C$ or $\mathbb R$. What conditions should be imposed on a topological space that it was true that a $k$-dimensional vector bundle defined over $F$ on ...
8
votes
1answer
1k views

The preimage of a maximal ideal is maximal

I'm having some difficulty with this homework problem: Let $A, B$ be reduced finitely generated $\mathbb{C}$-algebras, and $\psi : A \longrightarrow B$ a $\mathbb{C}$-algebra homomorphism. Let ...
7
votes
1answer
345 views

Gluing sheaves - can we realize $\mathcal{F}(W)$ as some kind of limit?

Let $X$ be a topological space and $\{U_i\}$ and open cover for $X$. Suppose we have sheaves $\mathcal{F}_i$ on $U_i$ and for each $i,j$ an isomorphism $\varphi_{ij} : \mathcal{F}_i|_{U_i \cap U_j} ...
7
votes
1answer
526 views

Meaning of holomorphic Euler characteristics?

I wonder what holomorphic Euler characteristic $\chi(\mathcal{O}_X)$ of a variety represents. For example, I have seen someone fix $\chi(\mathcal{O}_C)=n$ for a complex curve $C$. What does this mean ...
7
votes
1answer
198 views

the fundamental exact sequence associated to a closed space

Let $(X,\mathcal O_X)$ be an algebraic variety. If $Y\subseteq X$ is a closed subset, then we can equip $Y$ with a structure of algebraic variety $(Y,\mathcal O_Y)$. The function $i:Y\rightarrow X$ ...
6
votes
1answer
217 views

Cardinality and degrees of irreducible components of an affine variety

Let $Q_1,\ldots,Q_s \in k[x_1,\ldots,x_n]$, where $k$ is not necessarily algebraically closed (I'm thinking of $k$ as some field with positive characteristic $p$). I'm somewhat new to the world of ...
6
votes
2answers
260 views

For what algebraic curves do rational points form a group?

For what real algebraic curves do rational points form a group ? How does this relate to Jacobian Varieties ?
3
votes
2answers
244 views

Sites or youtube videos to learn algebraic geometry

Is there any sites or free lecture videos to learn algebraic geometry? or should I call abstract algebra? I want to understand about rings, ideals, and real spectrum of rings but my understanding on ...
3
votes
3answers
345 views

What is the support of a localised module?

Let $R$ be a noetherian commutative ring, and let $\mathfrak{m}$ be a maximal ideal of $R$. Let $M$ be a finitely-generated torsion $R_\mathfrak{m}$-module, considered as an $R$-module. Is it possible ...
19
votes
1answer
505 views

What is $\operatorname{Spec}\mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$?

Let $X = \operatorname{Spec} \mathbf{C}[[x,y]]/(y^{2} - x^{3} - x^{2})$. I would like to describe $X$ set-theoretically. My questions are: Can one explicitly say what the elements in $X$ are? Is it ...
12
votes
3answers
1k views

Why is there no polynomial parametrization for the circle?

How does one show that the unit circle admits no polynomial parametrization? What is needed for this, are there general criteria? Thanks
10
votes
2answers
422 views

Variety of Nilpotent Matrices

Let $k$ be an algebraically closed field and view $M_n(k)$ as $\mathbb{A}^{n^2}$. $A\in M_n(k)$ is nilpotent if and only if $A^n=0$. Since the equation $A^n=0$ is given by $n^2$ polynomial ...
10
votes
2answers
318 views

Is a divisor in the hyperplane class necessarily a hyperplane divisor?

Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$. ...
9
votes
4answers
608 views

Spectrum of $\mathbb{Z}[x]$

Can someone point me towards a resource that proves that the spectrum of $\mathbb{Z}[x]$ consists of ideals $(p,f)$ where $p$ prime or zero and $f$ irred mod $p$? In particular I remember this can be ...
8
votes
1answer
324 views

Properties of quotient sheaves

I am reading Hartshorne's Algebraic Geometry, II.6 about Cartier Divisor. It is defined to be the global section of the sheaf $K^*/O^*$. Then it said: " thinking of the properties of the quotient ...
8
votes
0answers
205 views

Ideal of the pullback of a closed subscheme

Let $f : X \to Y$ be a morphism of schemes and $J \subseteq \mathcal{O}_Y$ a quasi-coherent ideal. Let $I$ denote the image of $f^* J \to f^* \mathcal{O}_Y = \mathcal{O}_X$. Then $I \subseteq ...
7
votes
2answers
338 views

How to think of the Zariski tangent space

The Zariski tangent space at a point $\mathfrak m$ is defined as the dual of $\mathfrak m/\mathfrak m ^2$. While I do appreciate this definition, I find it hard to work with, because we are not given ...
7
votes
1answer
192 views

Why is the kernel of this strange polynomial homomorphism what it is?

I've been trying to delve a little further into linear algebra, but I'm not following something I think is supposed to be obvious. Suppose $M_{m,n}(\mathbb{C})$ is the set of rectangular $m\times n$ ...
7
votes
2answers
763 views

Chern numbers of Projective Space

Consider the $k$-th chern class $c_k:=c_k(\mathcal{T}_{\mathbb{P}^n})$ of the tangent sheaf of projective space $\mathbb{P}^n=\mathbb{P}^n_\Bbbk$ over some (algebraically closed, if you want) field ...
7
votes
4answers
486 views

Determining the generators of $I(X)$

Let $ X \subset \mathbb A^3 $ be the union of three coordinate axes. How do I determine the generators of $I(X)$? Also, how do I show it has at least 3 elements?
7
votes
2answers
506 views

Elementary question about Cayley Hamilton theorem and Zariski topology

A question about a proof of Cayley's Hamilton theorem using Zariski topology. "The set $C$ of all matrices of size $n \times n$ (over an algebraically closed field $k$) with distinct eigenvalues is ...
6
votes
2answers
150 views

On the $j_!$ of a sheaf

Let $X$ be a topological space, and $U$ an open subset. Denote $j:U\to X$ the inclusion. Let $\mathcal F$ be a sheaf on $U$. We define $j_!\mathcal F$ to be the sheaf associated to the presheaf ...
6
votes
2answers
191 views

Is the circle a rational curve and what is its function field?

It does seem like the circle ($S^1=\{X^2+Y^2=1\}\subseteq k^2$ for $k$ a field) is a rational curve: it has parameterization $X=2T/(T^2+1)$ and $Y=(T^2-1)/(T^2+1)$. On the other hand, we have a ...
6
votes
4answers
522 views

Reference request: Vector bundles and line bundles etc.

I am interested in learning algebraic geometry and I talked to one of my professors today (who is, in part, an algebraic geometer) and he recommended I understand the analytic analogue of the ideas in ...
6
votes
1answer
2k views

Picard group and cohomology

It's an easy but boring exercise (Hartshorne Ex. III.4.5 or Liu 5.2.7) that the group $Pic(X)$ of isomorphism classes of invertible sheaves on a ringed topological space (well, maybe we can restrict ...
6
votes
2answers
1k views

Singular and Sheaf Cohomology

Let $X$ be a complex manifold of dimension $n$. Thus, it's a real manifold of dimension $2n$. Now cohomology is a topological concept so it should not depend upon the structure given on a topological ...
5
votes
0answers
109 views

Configuration scheme of $n$ points

If $X$ is a space, the configuration space of $n$ (distinct) points in $X$ is $C_n(X)=F_n(X)/\Sigma_n$, where $F_n(X) = \{x \in X^n : \forall i,j (i \neq j \Rightarrow x_i \neq x_j)\}$ is the ...
5
votes
2answers
186 views

If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?

Let $A$ be a graded ring. Note that the grading of $A$ may not be $\mathbb{N}$, for example, the grading of $A$ could be $\mathbb{Z}^n$. Actually, my question comes from the paper of Tamafumi's On ...
5
votes
1answer
181 views

$X \to Y$ flat $\Rightarrow$ the image of a closed point is also a closed point?

This question came from a proof in Algebraic Geometry by Hartshorne (Chapt3, Corollary 9.6) To be precise, Let $f:X \to Y$ be a flat morphism of schemes of finite type over a field $k$. Then is it ...
5
votes
3answers
233 views

Can every element in the stalk be represented by a section in the top space?

Let $S$ be a sheaf over $X$ and $r$ an element in $S_x$ for some $x$ in $X$. Must there exist a section $s$ in $S(X)$ such that such that $s$ equals $r$ when mapped to $S_x$ by the canonical map?
4
votes
0answers
74 views

Computing an integral basis of an algebraic function field, $y^4-2zy^2+z^2-z^4-z^3=0$.

I am trying to compute an integral basis for the algebraic extension $K(z,y)$ of $K(z)$ by $y$, with $f(z,y)=0$, $$ f(z,y) = y^4-2zy^2+z^2-z^4-z^3 = 0. $$ $K$ here is either $\mathbb{Q}$ or ...
4
votes
1answer
208 views

Pullback of maximal ideal in $k[y]$ is not maximal in $k[x]$.

Let $k$ be a field and let $k[x]=k[x_1,\ldots,k_m]$ and $k[y]=k[y_1,\ldots,y_n]$ be $k$-algebras. Let $\varphi:k[x]\to k[y]$ be a $k$-algebra homomorphism. It can be shown (pretty readily) that ...
4
votes
3answers
999 views

Ideal of the twisted cubic

The twisted cubic is the image of the morphism $\phi : \mathbb{P}^1 \to \mathbb{P}^3 , (x:y) \mapsto (x^3:x^2 y:x y^2:y^3)$, it is given by $X = V(ad-bc,b^2-ac,c^2-bd)$. Now I would like to compute ...
4
votes
5answers
328 views

Derived category and so on

I am looking for an introductive reference to the theory of derived categories. Especially I need to start from the very beginning and I need to know how to use this in examples which comes from ...
3
votes
1answer
165 views

Closure of image of diagonal morphism of S-scheme

Let $X$ be an $S$-scheme with structural morphism given by $f : X \to S$. The image of the diagonal morphism $\Delta : X \to X \times_S X$ is contained in the subset $Z := \{ z \in X \times_S X : ...
3
votes
1answer
173 views

Exact sequence in Beauville's “Complex Algebraic Surfaces”

On page 3 of Beauville's book (Lemma I.5) he takes two curves $C$ and $C'$ in a surface $S$ an takes global sections $s\in H^0(S,\mathcal{O}_S(C))$ and $s'\in H^0(S,\mathcal{O}_S(C'))$. In a recent ...
3
votes
1answer
169 views

$M_n\cong\Gamma(\operatorname{Proj}S.,\widetilde{M(n).})$ for sufficiently large $n$

Let $S.$ be a graded ring, finitely generated by degree 1 elements as a $S_0$-algebra. Let $M.$ be a finitely generated graded $S.$-module. There exists a natrual map ...
1
vote
1answer
72 views

Reconstruct shape of a body from rationality of its projections

There is a closed convex body $S$ in $\mathbb{R}^3$. Areas of its projections on all planes (not only those normal to axes $x,y,z$) are rational numbers. Can we deduce that $S$ is a ball? Replace ...
13
votes
3answers
681 views

Localization at a prime ideal is a reduced ring

Here is the question that I came up with, which I am having trouble proving or disproving: Let $A$ be a ring (commutative). Let $p \in Spec(A)$ such that $A_p$ is reduced. Then there exists an open ...
11
votes
1answer
229 views

What is the best way to see that the dimension of the moduli space of curves of genus $g>1$ is $3g-3$?

This fact was apparently known to Riemann. How did Riemann think about this?
11
votes
2answers
852 views

Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$

I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me ...
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 ...
8
votes
1answer
181 views

Hom between 2 schemes

Why is the set $Hom(X,Y)$ between 2 schemes $X$ and $Y$ a scheme as well? Where can I read the construction? For example, $Hom(\mathbb{A}^1,\mathbb{A}^1)$ is the set of all polynomial, and what is the ...
7
votes
1answer
114 views

How to tell whether a scheme is reduced from its functor of points?

Suppose I have a scheme $X$ and I want to know if $X$ is reduced, but all I have access to is the functor $$ R\mapsto X(R)=Mor(\operatorname{Spec}(R),X) $$ from commutative rings to sets (rather ...
7
votes
0answers
475 views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
7
votes
2answers
270 views

Is this quotient ring $\mathbb{C}[z_{ij}]/\ker\phi$ integrally closed?

A few days ago, I asked a linear algebra question, but it seems that the notions are better stated in terms of algebraic geometry. I don't have much solid knowledge of algebraic geometry, so I'm ...
7
votes
1answer
249 views

Construction of the global $\mathbf{Proj}$

I have many questions about the very abstract concept of global $\mathbf{Proj}$. I am following Hartshorne's book Algebraic Geometry, where this concept is on II.7, page 160. Let $(X, ...
7
votes
2answers
540 views

Geometric reason why elliptic curve group law is associative

The question title says it all. I am looking for a geometric proof for the fact that the group law defined on elliptic curves is associative. I've heard somewhere about something on the ...