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. ...

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4
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0answers
26 views

The complete solution to a system of polynomials over $\mathbb{R}$

If I am solving a positive-dimensional system of polynomials over $\mathbb{R}$, and specifically am searching only for real solutions, how do I know that my solution is complete and there are no other ...
1
vote
1answer
60 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
19 views

How do you define the restriction of a sheaf?

Just to be clear with the notations: Recall that the pullback of $\mathcal{F}\in\mathcal{O}_B\text{-Mod}$ via $f:A\rightarrow B$ (morphism of schemes) is defined as ...
-4
votes
1answer
26 views

I want to know the difference between Greatest and least element in Maths [on hold]

Please diggerentiate between Maximal/Minimal element and Greatest and least element in Discrete Mathematics.. Please help me
0
votes
0answers
19 views

Bounding the degree of an algebraic extension containing solutions to polynomials

Let $F$ be a field, and let $f_{1},\ldots, f_{s}$ be polynomials in $F[x_{1},\ldots, x_{t}]$. Assume that the degree of the polynomials is bounded by $d$, by which I mean, if $m$ is any term in ...
5
votes
1answer
50 views

Why do subvarieties correspond to Hodge classes?

Let $X$ be a smooth complex projective variety and define $$Hdg^k(X)=H^{2k}(X,\mathbb{Z})\cap H^{k,k}(X)$$ the group of integral $(k,k)$ cycles on $X$. Now it is a fact that we can associate to the ...
0
votes
1answer
33 views

Conics and conics of the form $ax^2+by^2+c=0$

The problem of finding rational points on conics is usually discussed (for example in the book of Silverman and Tate) for conics of the form $ax^2+by^2+c=0$. I assume that those conics are in ...
1
vote
0answers
23 views

Does an isogeny always define a covering map?

Consider a map $f: G_1 \to G_2$ between two topological groups. If $f$ is an isogeny when viewing $G_1,G_2$ as algebraic groups does $f$ always define a covering map when viewing $G_1,G_2$ as ...
1
vote
1answer
47 views

Understanding a proof of Hartshorne's book proposition 2.2.

I have been reading the book Algebraic Geometry by Robin Hartshorne and I have found the following proposition: For part b) the proof goes as follows: The thing is that, How can we ...
0
votes
1answer
19 views

Help in showing that the cusp $(y^2-x^3)\subset \mathbb{C}^2$ is not isomorphic to $\mathbb{C}$

Let $X:=(y^2-x^3)\subset \mathbb{C}^2$ be the vanishing of the polynomial $f(x,y)=y^2-x^3.$ I have proved an exercise in Hartshorne: If $\varphi:\mathbb{C} \to X, \ t \mapsto (t^2,t^3)$ is the ...
1
vote
1answer
43 views

Mori cone and birational geometry

Let $X$ be a projective and smooth algebraic variety (maybe here the hypotheses may be relaxed). If I understand correctly, Mori cone is defined as the closure of the cone in $N_1(X)$ of effective ...
0
votes
0answers
18 views

Intersection of a variety with its tangent space

If you have a complete intersection of quadrics, what can be said about the dimension of its intersection with a general tangent space? Do you get the expected dimension? In my particular situation, I ...
0
votes
1answer
15 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
36 views

Is the dimension of an abelian variety always finite?

Is the dimension of an abelian variety always finite? If yes, why? With "Abelian variety" I mean a integral scheme $X$, proper over an algebraically closed field (complete variety) with a group law ...
-1
votes
1answer
14 views

Limit through a figure

If a circular arc of radius 1 subtends an angle of x radians . The centre of the circle is o and the point c is the intersection of two tangents lines at a and b . Now let T(x) be the area of the ...
3
votes
0answers
28 views

Explicit unit/counit of inverse image/direct image adjunction.

Is there a nice explicit description for the unit and counit of the inverse image/direct image adjunction $f^{-1} \dashv f_*$ between sheaves of rings (and in the version $f^* \dashv f_*$ for ...
2
votes
0answers
19 views

Dimension of an abelian subvariety in the proof of the Poincaré's Reducibility Theorem

I am trying to understand the proof of the Poincaré's Reducibility Theorem, that I'm reading from the book "Abelian varieties" of J.S. Milne (see Proposition 10.1) and from the book "Abelian ...
0
votes
0answers
26 views

Grassmannian as schemes

I would like to understand the Grassmannian as a scheme. If $V$ is a vector space over the complex numbers, then $\mathbb{C}$-valued points of the Grassmannian $\mathbf{Grass}(r,V)$ consists of all ...
1
vote
1answer
29 views

Reference for the $\Bbb A^1_k$-rigidity of abelian $k$-varieties

Is there a reference that shows, for a field $k$, that abelian $k$-varieties are $\Bbb A^1_k$-rigid? A smooth variety $X$ over $k$ is $\Bbb A^1_k$-rigid if and only if the canonical map $$ ...
0
votes
0answers
35 views

Dimension of the zero-th cohomology for an ample line bundle [on hold]

Given an ample line bundle L on an abelian variety X, we have that $\dim H^0(X,L)>0$. Why?
2
votes
0answers
49 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
20 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
25 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 ...
-1
votes
0answers
31 views

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

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 $\mathcal{F}(U\cup V)=\mathcal{F}(U)\times \mathcal{F}(V)$ ...
0
votes
0answers
18 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
30 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
29 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
13 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
24 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
22 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 ...
3
votes
0answers
25 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
17 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
163 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
35 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 ...
1
vote
1answer
21 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 ...
21
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
42 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
91 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
21 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
37 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 $|L|$ ?
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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.
2
votes
0answers
38 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), ...
0
votes
0answers
51 views
2
votes
1answer
48 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
31 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
37 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
27 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
36 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 ...