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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3
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23 views

Computing Hodge numbers of a complete intersection

The situation is this: I have a 5-dimensional irreducible projective variety $Y$ embedded in $\mathbb P^{13}$. This variety is singular, the singularities being a disjoint union of two curves. I have ...
2
votes
0answers
17 views

Splitting varieties of two Galois cohomology symbols

One important property of the so called norm varieties defined by Rost, is that they split pure symbols of Galois cohomology groups $H^n(k,\mu_p)$: For some $\alpha \in H^n(k,\mu_p)$ there is a ...
0
votes
0answers
20 views

Space formed by difference between lines

I'm trying to write a program that takes in two lines in parametric form (an offset point and direction vector) and a constant number. It outputs pairs of s and t (the magnitude of the lines' ...
0
votes
0answers
20 views

Identity of dimensions related to cohomology group of projective space

Let $k$ be a field, $S=k[T_0,\ldots T_r]$, $\mathbb P=\mathbb P_k^r=\operatorname{Proj}S$ and $O$ a structure sheaf of $\mathbb P$. How can I show the identity $$\operatorname{dim}_kH^0(\mathbb ...
1
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0answers
21 views

A question about intersection number on surfaces

This question is from the Qing Liu's book: Algebraic Geometry and Arithmetic Curves, Exercise 9.1.6. Let $X\to S$ be an arithmetic surface and $X_s$ a closed fiber. Let $C_1,...,C_m$ denote the ...
1
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0answers
12 views

Solvable algebraic groups and base-change

Let $G$ be an algebraic group over a field $k$ of characteristic zero. My understanding is that solvability is not a geometric property (is this correct though?). This motivates my questions: Let ...
0
votes
1answer
24 views

Clarifications about parabolic subgroups of $GL_4$

I'm asked to find all the parabolic groups $P$ which contains $T_4$, the subgroup (borel) of upper triangular invertible matrices. This is my definition of parabolic subgroup Let $P$ be a subgroup ...
0
votes
0answers
21 views

Understanding a theorem of Saint-Donat

In his thesis on K3 surfaces Saint-Donat proves the following fact (thm 6.1) Let $L$ be a line bundle on a K3 surface $X$ such that the linear system $|L|$ has no fixed components and the morphism ...
2
votes
1answer
44 views

Morphism of locally free $\mathcal{O}_X$-modules and projection

Let $X$ be a proper smooth variety over a finite field. Let $\mathcal{F}$ be a locally free $\mathcal{O}_X$-module and $\psi:\mathcal{O}_X\rightarrow\mathcal{F}$ be an injection and locally splitting. ...
3
votes
1answer
42 views

What's the relation between Galois theory of fields and algebraic varieties?

Galois theory of fields gives correspondence between closed subgroups of Galois group and closed intermediate fields of the extension. Elementary algebraic geometry gives correspondence between ...
1
vote
1answer
41 views

$k$-point after base change

If $X$ is a variety over $k$, is it true that there exists a finite separable extension $k'$ of $k$ such that $X$ has a $k'$-point? What if we can assume $X$ is a smooth projective curve? This seems ...
1
vote
1answer
29 views

Existence of Hilbert's polynomial

I heard that Hilbert's syzygy theorem can be used to show the existence of Hilbert polynomials. How does the construction works? Namely, why do every coherent $O$-module $\mathscr F$ the ...
4
votes
1answer
49 views

Open Set of Non-zero Divisors of a Module

Let $R=k[x_1,\dots,x_r]$ be the polynomial ring over the field $k$. Denote by $R_1$ the vector space of linear forms, i.e. all the degree-$1$ elements of $R$. Let $M \neq 0$ be a finitely generated ...
1
vote
1answer
26 views

What is the domain of definition of $S_1/S_0$ on $\mathbb{P}^2$?

Consider the regular function given by $S_1/S_0$ on the projective sphere $\mathbb{P}^2$ over a field $k$ (We can assume algebraically closed, if it's needed for some reason). I'm just worried, is the ...
-1
votes
1answer
16 views

Find the equation of a line in 3D space given elevation and azimuthal angles [on hold]

How do I find the equation of the 3D line that passes through the point 0,0,0 with an elevation angle of 57 degrees, and an angle of 207, between the X and y axes?
2
votes
1answer
33 views

Coherent modules and cohomology group

Let $k$ be a field, $S=k[T_0,\ldots,T_r]$, $\mathbb P=\mathbb P_k^r=\operatorname{Proj}S$ and $O$ the structure sheaf of $\mathbb P$. Let $\mathscr F$ be a coherent $O$-module. Why do the is an ...
2
votes
1answer
33 views

Sheaf on abelian variety preserved by tensor product with a translation invariant line bundle

If $A$ is an abelian variety, $\mathscr{F}$ is a coherent sheaf on $A$, and $\mathscr{F}\otimes \mathscr{L}\cong \mathscr{F}$ for all translation invariant line bundles $\mathscr{L}$, why is the ...
2
votes
0answers
67 views

Solving systems of polynomials with an oracle

I need to solve a system of polynomials. Let the variables be $x_1, \dots, x_n$, and let the polynomials be $f_1, \dots, f_n$ Let's say we have these conditions we can already assume: there are ...
1
vote
1answer
44 views

Is the contraction morphism proper?

This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up. Let $X$ be a projective variety and $F\in \overline{NE}(X)$ an ...
5
votes
1answer
52 views

Question about the construction of Mukai flop

Let X be a symplectic complex manifold of dimension $2n $, i.e. there exists a non degenerate holomorphic 2-form $\sigma $ such that $ H^0 (X,\Omega^2)=\mathbb {C}\cdot\sigma $. Suppose that there ...
0
votes
2answers
47 views

A question about the geometric representation of Spec $\Bbb{C}[x,y]/(x-y)$

The representation of Spec $\Bbb{C}[x,y]/(x-y)$ is given geometrically as the line $x-y=0$. I don't understand how this is. Can every prime ideal be represented as a point on the line $x-y=0$?
2
votes
1answer
66 views

Degree of a torsion-free subsheaf

Suppose that $R$ is a torsion-free subsheaf (of positive rank) in another torsion-free sheaf $S$, on a smooth complex projective variety $X$. If $S$ is (slope) semistable, is it true that the degree ...
0
votes
1answer
47 views

Why are minimal irreducible closed sets in $A^n$ single points?

In Hartshorne's Algebraic Geometry example 1.4.4, he says A maximal ideal $m$ of $A = k[x_1,\cdots,x_n]$ corresponds to a minimal irreducible closed subset of $A^n$, which must be a point ... I ...
1
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0answers
47 views

Algebraic Geometry Dealing With Elliptic Curves

Prove that there exists an infinite set of points, $$\dots, P_{-1}, P_0, P_1, P_2, P_3, \dots$$ in the plane with the following property: For any distinct three integers, say $a, b, c$, points $P_a, ...
0
votes
0answers
30 views

Regular map $\varphi\colon X\to Y$ is an isomorphism iff it's a homeomorphism and induces an isomorphism on local rings?

I was reading the following, but part is still unclear to me. Suppose $\varphi\colon X\to Y$ is a regular map of quasi-projective varieties. We get an induced map on local rings ...
3
votes
0answers
52 views

Analogue of the Hecke operator for a general curve

Background: The eigenvalues of the Hecke opeator $T_p$ on the space of cusp forms $S_2(\Gamma_0(N))$ (of dimension $g=\text{genus}(X_0(N))$) are analysed using reduction mod $p$ and the use of the ...
2
votes
1answer
34 views

How can we be sure that stalks are in the sheaf codomain?

A stalk is a colimit of all $\mathcal{F}(U)$ over all open sets $U$ containing $p$. A stalk is alternatively thought of as the collection of germs at a point $p$. Hence, it cannot be thought of ...
8
votes
0answers
65 views

What's the intution behind defining the cotangent sheaf as $\Delta^\ast(\mathscr{I}/\mathscr{I}^2)$?

This definition seems to be given all over the place (e.g. Hartshorne II.8, Vakil 21.2.20, Wikipedia, McKernan's lecture notes from MIT), and never with any explanation as to why the map $\Delta : X ...
0
votes
1answer
19 views

Nondegenerate Conics

I am having trouble seeing why the set of all nondegenerate conics is Zariski open in the parameter space $\mathbb{P}^5$.This is an exercise in Smith 5.2.2 that I am trying to do. I will be grateful ...
1
vote
0answers
27 views

General form of regular maps $\mathbb{P}^n\to\mathbb{P}^m$?

I'm reading through Milne's algebraic geometry notes, and there's a remark without justification I'm having trouble seeing. Essentially: Suppose $F : \mathbb{P}^n\to\mathbb{P}^m$ is a regular map ...
4
votes
3answers
112 views

Closed immersions are stable under base change

My question can be summarized as: I want to prove that closed immersions are stable under base change. This is exercise II.3.11.a in Hartshorne's Algebraic Geometry. I researched this for about ...
6
votes
0answers
53 views

Explaining the difference between the number theoretic Langlands program and geometric Langlands program to a graduate student.

I am a graduate student who just took a course introducing some notions in algebraic number theory and algebraic geometry (officially, it was a course on an introduction to the Langlands program). ...
1
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0answers
32 views

Maximal solvable subgroup not Borel

Given $G$ connected linear algebraic group I want to find a $U \subset G$ maximal solvable subgroup which is not connected. My attempt is to take $G=SO(n)$ for $n \geq 3$ and $U=D_{n}\cap SO(n)$: ...
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0answers
27 views

An unclear passage in a proof of a theorem about complex surfaces.

Look at the following theorem (I have posted only part of the proof): I don't understand the highlighted sentence. Why the morphism $\widetilde{p}:\widetilde X\longrightarrow C$ can't map a ...
1
vote
1answer
76 views
+50

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 ...
1
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0answers
73 views
+100

The points are $\mathbb{Z}$-linearly dependent

If $E/\mathbb{Q}$ the elliptic curve $y^2=x^3+x^2-25x+29$ and $$P_1=\left (\frac{61}{4}, \frac{-469}{8}\right ), P_2=\left ( \frac{-335}{81}, \frac{-6868}{729}\right ) , P_3=\left ( 21, 96\right )$$ ...
0
votes
0answers
27 views

Action of connected linear group on quasi projective variety with finitely many orbits

I'm asked to prove the following: Be $G$ connected linear algebraic group acting on $X$ quasi projective variety with finitely many orbits, show that every irreducible $G$-invariant subset $Y \subset ...
1
vote
0answers
40 views

Question on Grassmannian

Let $V \subset \mathbb{P}^n$ open, and $W \subset \operatorname{Grass}_d(\mathbb{P}^n)$ be the set of linear subspaces which are contained in $V$. Then the question is whether $W$ is open in ...
1
vote
0answers
72 views

The sum of three collinear points is zero.

Could you give me an idea how we could prove that at an elleiptic curve $E/ \mathbb{Q}$, the sum of three collinear rational points of it is equal to $0$, exactly then when the identity element of ...
1
vote
1answer
74 views
+100

Elliptic curve- Component of point

If we have $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$,then if $P_1=P_2$ we have that $$\lambda=\frac{3x_1^2+2a_2x_1+a_4-a_1y}{2y_1+a_1x_1+a_3}, v=\frac{-x_1^3+a_4x_1+2a_6-2_3y_1}{2y_1+a_1x_1+a_3}$$ and then ...
2
votes
1answer
45 views

What is the ideal of a point in algebraic geometry?

I found a problem as follows: Find the ideal of a point $z$, denoted by $\mathfrak j_z\subset\mathbb Q[X,Y]$, and its conjugates in $\mathbb C^2$ as $z=(\sqrt{2},\sqrt{3})$. I tried to Google but ...
0
votes
0answers
31 views

Meaning of “local equation” of a divisor.

Let $X$ be a smooth surface and moreover let $C,D$ be two effective divisors of $X$. Hartshorne says (page 357) that $C$ and $D$ meet transversally if the local equations $f,g$ of $C,D$ at $P$ ...
1
vote
0answers
62 views

Help with the proof of Max Noether's Residue Theorem from Fulton's book

I'm having problems understanding one part of the proof of the Residue Theorem, on chapter 8 of Fulton's book Algebraic Curves, section 8.1 (http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf page ...
2
votes
1answer
43 views

Etale morphism and reduced schemes

Let $f:X \to Y$ be an etale morphism of Noetherian schemes. Is it true that the induced morphism on the reduced schemes, i.e., $f_{\mathrm{red}}:X_{\mathrm{red}} \to Y_{\mathrm{red}}$ is etale as ...
7
votes
0answers
65 views

Non-Universal Delta Functors

Recall a delta functor on an abelian category is a collection of functors $H^n$, $n \ge 0$ and connecting homomorphisms associated to each short exact sequence so that we get a corresponding long ...
0
votes
1answer
38 views

The equation of a 3D surface bounded by 3 known elliptical curves

I am trying to find the equation of a 3D surface as illustrated below. The boundaries of this surface is comprised of two planar elliptical arcs $AB$ and $AC$ as well as a 3D arc $BC$ which is a 3D ...
2
votes
0answers
43 views

Canonical algebra isomorphism $k[D(f)]\cong k[S_0,\dots,S_n]_{(f)}$?

Here's a common set up. Suppose you have $f\in k[S_0,S_1,\dots,S_n]$ is a homogeneous polynomial with $\deg(f)=d$, over some closed field $k$. Let $D(f)$ be the principal open set of $f$ in projective ...
1
vote
0answers
33 views

Eisenbud/Haris, Exercice I-53: morphism between global spectra

Let $X=\operatorname{Spec}(\mathscr{F})$ and $Y=\operatorname{Spec}(\mathscr{G})$ two global spectra over a scheme $S$ and let $f:X\to Y$ be a morphism. I want to show that for all prime ideal sheaf ...
3
votes
0answers
60 views

Evaluate a rational function at infinity

In the context of the Tate pairing, I would like to know that it means to `evaluate' an $\mathbb{F}_{q^k}$-rational function at $\infty$. For instance, the reduced Tate pairing is $e_n:G_1\times ...
1
vote
0answers
46 views

Question related to the definition of affine schemes

The definition for affine schemes I have learned was that it is a locally ringed space $(X, O_X)$ that is isomorphic to $(Spec \ A, O_{ Spec \ A })$ for some ring $A$ (commutative and includes $1$). ...