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)

2
votes
0answers
41 views
+50

Where can I learn about differential graded algebras?

I want to learn more about differential graded algebras so that I can construct explicit examples of derived schemes over characteristic 0, compute smooth resolutions of morphisms of schemes, and ...
1
vote
0answers
17 views

Polynomial division for identifying an expression in terms of complex numbers.

This question is blatantly copied from here, for the sake of learning more I specify it a bit more: $$f(z)= (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y)$$ $$z = a+bi$$ I want to write $f(z)$ in terms of ...
1
vote
0answers
14 views

Formula relating dimension of fiber of morphism between varieties

Let $f: X \to Y$ be a morphism of (irreducible) varieties, where the dimension of every fiber dim$f^{-1}(y)=n$ is the same. Must it follow that dim$X=$ dim$Y+n$? The reason I am asking this is that ...
0
votes
0answers
17 views

If $U$ and $V$ are disjoint, then why is $\mathcal{F}(U\cup V)=\mathcal{F}(U)\times \mathcal{F}(V)$?

Let $X$ be a topological space such that $U,V$ are disjoint open sets on it. Let $\mathcal{F}$ be a sheaf on $X$. Then Vakil's notes say that $\mathcal{F}(U\cup V)=\mathcal{F}(U)\times \mathcal{F}(V)$ ...
1
vote
0answers
21 views

normalization of a curve, simplest example

I am learning about normalization of nodal curves and I am trying to understand the simplest example: $xy=0$ As far as I understand its coordinate ring is $k[x]\oplus k[y]$ (let $k$ be an ...
0
votes
0answers
12 views

The diagonal in a product of any variety with itself is a subvariety?

I'm working on Exercise 2.15 in Algebraic Geometry: A First Course by Joe Harris. The first part of the exercise (Show that the image of the diagonal $\Delta \subset \mathbb{P}^n \times \mathbb{P}^n$ ...
0
votes
0answers
26 views

Segre embedding $\mathbb P^1\times\mathbb P^1\to\mathbb P^3$

Let $\Psi:\mathbb P^1\times\mathbb P^1\to\mathbb P^3$ be the map $$((x_0:x_1),(y_0:y_1))\mapsto (x_0y_0:x_0y_1:x_1y_0:x_1y_1)$$ and let $Q$ be the image of $\Psi$. I have shown that $Q$ is the zero ...
2
votes
1answer
24 views

Let $G$ be an abelian group can we always construct a quasi projective variety $X$ such that Cl$(X)=G$

Recall that for a quasi projective variety $X$ one can define the Divisor Class Group denoted by Cl$(X)$ Let $G$ be an abelian group. Can we always construct a quasi projective variety $X$ such ...
0
votes
0answers
10 views

Verifying that the weighted projective space $\Bbb{P}_{\Bbb{Q}}(1,1,2,3)$ is singular.

I while ago I attended a talk that was somewhat over my head, and the speaker mentioned in passing that the weighted projective space $\Bbb{P}:=\Bbb{P}_{\Bbb{Q}}(1,1,2,3)$ is singular. I suppose this ...
0
votes
0answers
22 views

What are the linear equivalence relations being talked about in the definition of a Picard group?

These set of notes say the following: The set of divisors Div$(X)$ on a compact Reimann surface $X$, modulo linear equivalence relations, forms a group Pic$(X)$. What are these "linear ...
1
vote
0answers
17 views

Example of two affine varieties $X,Y$ such that the image of $\phi:X \rightarrow Y$ is not locally closed

In my course Algebraic Geometry I always find it hard to come up with examples or counterexamples. For instance in the following question: Give an example of two affine varieties $X,Y$ and a morphism ...
2
votes
0answers
22 views

Arithmetic genus of divisors on cubic surface

This is a question from Hartshorne's Algebraic Geometry (Chapter V.4). It asks to show that for any divisor $D$ of degree $d$ on a smooth cubic surface $X$ in $\mathbb{P}^3$ the following inequality ...
0
votes
0answers
10 views

Weyl group, bilinear form, and character/cocharacter pairing. Many questions!

Let $G$ be a connected linear algebraic group, $T$ a maximal torus of $G$, and $\alpha$ a weight of $T$ such that $G_{\alpha} = Z_G(S)$ is not solvable, where $S = (\textrm{Ker } \alpha)^0$. I have ...
-1
votes
0answers
156 views
+50

How is this fact implicated?

Let $K$ be a field and $\overline{K}$ its algebraic closure, then we define the $n$-dimensional affine space as $$\mathbb{A}^{n}=\{(x_1, \ldots, x_{n})\mid x_1, \ldots, x_{n} \in \overline{K}\}$$ So ...
0
votes
2answers
32 views

is an open subset of a noetherian connected topological space connected?

Let $X$ be a noetherian connected topological space $X$. If $U$ is an open subset of $X$, must $U$ be connected? This is not true if $X$ is not noetherian e.g take $X=\mathbb{R}$ in the usual ...
0
votes
1answer
18 views

Homotopy group of the conformal group

I would like to know which are the first three homotopy groups of the conformal group SO(4,2): $$ \pi_n(SO(4,2))=? \quad n=1,2,3 $$
0
votes
1answer
36 views

Why does a linear equation define a point-set of dimension one less than the space?

For example: If we are in 2-space (2 unknowns), a linear equation defines a line. If we are in 3-space (3 unknowns), a linear equation defines a plane. I mean, it seems obvious, but an explanation ...
20
votes
3answers
1k views

Why was Sheaf cohomology invented?

Sheaf cohomology was first introduced into algebraic geometry by Serre. He used Čech cohomology to define sheaf cohomology. Grothendeick then later gave a more abstract definition of the right derived ...
1
vote
0answers
40 views

Associating to every line a vector in that line in an algebraic way

Let $X$ be a complex variety and $\mathscr E$ a locally free sheaf on $X$. Consider the fiber bundles $$ \mathbb P(\mathscr E) \overset{def}= \mathrm{Proj}(\mathrm{Sym}(\mathscr E)), \quad \mathbb ...
2
votes
3answers
90 views

4-ellipse with distance R from four foci

I'm trying to find the equation for the generalization of an ellipse called a $n$-ellipse which has a constant distance R from four foci located at $(0,0),(0,1),(1,0),(1,1)$ Edit: As an algebraic ...
0
votes
0answers
20 views

component of a divisor is mobile

If D is a mobile divisor, why there exists a divisor $D'\in |D|$, such that any component of $D'$ is mobile? Somebody told me that it follows from Bertini Theorem. Can anybody give me some details?
1
vote
2answers
88 views

Splitness of a short exact sequence on a curve

Let $C$ be a curve with genus $g > 1$. Consider the product $C \times C$, with natural projections $p_1$ and $p_2$ (from the first and second factor, respectively) to $C$. Consider the following ...
1
vote
0answers
35 views

general member of a linear system

Let $|L|$ be a linear system of a quasi protective variety X, and Z a subvariety of X. If Z is not contained in BS($|L|$), is it true that Z is not contained in a general member of $|L|$ ?
0
votes
0answers
11 views

Two collineations

Give collineations to prove the following (in the extended projective plane): a, One cannot contruct the midpoint of two points using a straightedge only. b, One cannot construct the reflection of a ...
0
votes
0answers
25 views

Is $A^2 - (0,0)$ quasicompact?

Is $\mathbb{A}^2 - (0,0)$ quasi-compact (in the Zariski topology)? Surely this is well known. I ask because then it would give an example of a quasi-affine scheme that is not affine.
1
vote
0answers
35 views

What *is* the coordinate ring K[V] of an algebraic variety V?

I've been trying to understand this for a while. If I understand it, we let V be an algebraic variety (set?) then define I(V) to be the ideal generated by V. The coordinate ring is K[V] = K[X]/I(V), ...
-1
votes
0answers
48 views
2
votes
1answer
47 views

Is a hypersurface really defined by an arbitrary polynomial?

In An Invitation to Algebraic Geometry Karen Smith writes at the beginning of the book: The zero set of a single polynomial in arbitrary dimension is called a hypersurface in $\mathbb C^n$. The ...
1
vote
0answers
29 views

How to fix this proof that isomorphic varieties have the same dimension? Is it possible?

I am trying to prove the following: Show that affine algebraic varieties that are isomorphic have the same dimension. For completeness let's state the definitions: Let $V,W$ be varieties. ...
0
votes
0answers
36 views

Solution verification: Curve is given by points $(t,t^2, t^3)$

I tried to solve the following exerice: Show that the twisted cubic curve corresponding to the affine variety $V(x^2 - y)\cap V(x^3 - z)$ consists of all points in $\mathbb A^3$ of the form ...
0
votes
0answers
25 views

How to find the Coordinate equation of a curve which bends all the parallel rays from infinity towards a single point

How should I proceed on to find the coordinate equation of a curve such that it bends all the parallel rays coming from infinity towards a single point. Yes I know that it would be a 2nd degree ...
0
votes
0answers
33 views

Picard number of Kahler manifold

Let $(M,\omega)$ be a Kahler manifold. How can we define simply the Picard number for the special case where $M$ is also projective? Wikipedia defines it as the rank of the Neron-Severi group. In ...
2
votes
0answers
42 views

adjoint functor of inverse image functor

$f: U\hookrightarrow X$ an open immersion of two complex manifolds. $f^{-1}$ is inverse image functor, in usual sense, from category of sheaves of abelian groups $\mathcal{Ab}(X)$ over $X$ to category ...
1
vote
1answer
62 views

the definition of “Birational Equivalence”

I have confused with the definition of "Birational Equivalence" in the Algebraic Geometry. In My Text book , ($V$ and $W$ are irreducible quasi-projective varieties) A rational map $f : V \to W$ is ...
1
vote
1answer
55 views

How to resolve the singularity of $xy+z^4=0$?

This singularity can not be resolved by one time blow-up. I don't know how to blow up the singularity of the "variety" obtained by the first blow-up, in other words, I am confused with how to do the ...
2
votes
1answer
43 views

morphism of sheaves on $\mathbb{R}/\mathbb{Z}$

Let $\mathscr{Z}$ be an arbitrary sheaf on $\mathbb{R}/\mathbb{Z}=X$ (with the quotient topology). Let $\mathscr{F}$ and $\mathscr{G}$ denote the sheaves of continuous functions on $X$ with values in ...
-1
votes
1answer
31 views

$k[t]$ is finitely generated $k[x,y]/(y^2-x^2-x^3)$ -module

I am reading example 3, section 7.3 of vakil's notes. It says that $k[t]$ is a finitely generated $k[x,y]/(y^2-x^2-x^3)$ -module by 1,t. This really confuses me. And he also claim $D(t^2-1)$ is ...
1
vote
0answers
29 views

sequence of cohomology groups associated to Koszul complex

In the paper "On branched coverings of some homogeneous spaces" of Kim and Manivel one reads, that there is a Koszul complex associated with a section $s$ of a locally free sheaf $S$ of rank $p$ (in ...
3
votes
2answers
229 views

Kernel of a morphism from a locally free sheaf is locally free

Let $C$ be a projective curve (not necessarily reduced or irreducible). Let $\mathcal{F}, \mathcal{G}$ be $\mathcal{O}_C$-modules and $\phi:\mathcal{F} \to \mathcal{G}$ be a morphism of ...
0
votes
1answer
24 views

Analytical isomorphism implies same multiplicities [duplicate]

I want to prove the following problem in Robin Hartshorne's Algebraic Geometry Chapter 1 exercise 5.14 If $P\in Y$ and $Q\in Z$ are analytically isomorphic plane curve singularities, show that the ...
3
votes
2answers
47 views

Exercise 1.11 Harris Algebraic Geometry: A First Course

I am trying to do part (b) of Exercise 1.11 in Harris' book Algebraic Geometry: A First Course. Let $F_0=Z_0Z_2−Z_1^2$, $F_1=Z_0Z_3−Z_1Z_2$, $F_2=Z_1Z_3−Z_2^2$ (s.t. $V(F_0,F_1,F_2)$ is the twisted ...
0
votes
0answers
20 views

"Correct'' morphism extension to Nagata compactifications

Can a morphism of separated schemes of finite type over a field be extended to Nagata compactifications of the schemes preserving the closed complements? Let $\mathbf{Sch}/k$ be the category of ...
1
vote
0answers
20 views

ADE classification of singular surfaces (catastrophe theory)

I have seen a lot the Arnold's classification of singular surfaces by the simple Lie groups. I have even asked the author of a book that used this classification about its origin and his answer was ...
0
votes
1answer
92 views

Plotting 3 equidistant points on a sphere

I'm trying to figure out how to plot with $x,y,z$, three points that are equidistant along the surface of a sphere from each other that are all on a horizontal axis (so $y = 0$) with a radius of $500$ ...
2
votes
1answer
38 views

Is there a projective morphism from the quadric surface to the projective plane with degree 1?

Is there a projective morphism from the quadric surface $\mathbb{P}^1\times\mathbb{P}^1$ to the projective plane $\mathbb{P}^2$, with degree $1$?
0
votes
1answer
67 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point ...
5
votes
1answer
72 views

Cohomology Class of a Subvariety

I'm working on question 7.4 of Chapter III.7 in Hartshorne's Algebraic Geometry. The question is about the cohomology class of a subvariety. The setup is as follows: $X$ is an $n$-dimensional ...
1
vote
1answer
68 views

Finiteness of Zeros and Poles on Noetherian schemes

This exercise comes from Ravi Vakil's notes. Suppose that $X$ is an integral Noetherian scheme, and $f \in K(X)^{\times }$ is a nonzero element of its function field. Show that $f$ has a finite number ...
1
vote
1answer
42 views

Exact sequence of sheaves and associated sequence of graded modules

Let $(X,\mathcal{O}_X)$ with $X=\mathbb{P}^n$ and consider a exact sequence of sheaves of $\mathcal{O}_X$-modules $$0 \to \mathcal{F} \to \mathcal{H} \to \mathcal{G} \to 0 $$ Suppose that we apply the ...
5
votes
2answers
123 views

Why are projective spaces and varieties preferable?

I am reading Hartshorne's Algebraic Geometry and it seems to me that projective spaces and varieties are prefferable. I don't know why. In a more elementary stage of mathematics, when we try to find ...