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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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 ...
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21 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 ...
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2answers
39 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 $x-y=0$?
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27 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 $X$. Is it true that the degree of $R$ is always less than or ...
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1answer
41 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 ...
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41 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, ...
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25 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 ...
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34 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 ...
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1answer
33 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 ...
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56 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 ...
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1answer
18 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 ...
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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 ...
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2answers
83 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 ...
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46 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). ...
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26 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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24 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 ...
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1answer
50 views

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 ...
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45 views

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

If $E/\mathbb{Q}$ the elliptic curve $y^2=x^3+x^2-25x+29$ and $$P_1=\left (\frac{61}{4}, \frac{-469}{8}\right ), P_2=\left ( \frac{-335}{81}, \frac{-6868}{729}\right ) , P_3=\left ( 21, 96\right )$$ ...
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23 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 ...
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39 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 ...
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53 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 ...
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1answer
48 views

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 ...
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1answer
44 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 ...
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29 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 $P$ ...
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59 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 ...
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1answer
41 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 ...
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61 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 ...
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1answer
37 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 ...
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42 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 ...
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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 ...
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57 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 ...
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45 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 $1$). ...
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1answer
27 views

How to calculate 2-d plane from 3 4-d points?

I want to compute 3-d cross-sections of a pentatope (4-dimensional tetrahedron). The 3-d cross-sections will be calculated as: x+y+z+w=c C is a constant that I will vary to get different ...
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2answers
71 views

Is there a classification of regular maps $\mathbb{P}^1(k)\to\mathbb{A}^1(k)$?

If $\mathbb{P}^1(k)$ and $\mathbb{A}^1(k)$ are the projective line and affine line, respectively, over an algebraically closed field $k$, is there any known classification of the regular maps ...
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2answers
72 views

Singular Points on Algebraic Curves

What does it mean for a point on an algebraic curve (over any field) to be singular? Over $\mathbb{R}$ or $\mathbb{C}$ is means that all derivatives vanish, but how does this definition generalise to ...
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0answers
39 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
2
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1answer
53 views

Base change for Quot-scheme

I am reading the book of Huybrechts and Lehn "The Geometry of Moduli Spaces of Sheaves" with an aim to become a little bit familiar with this topic. Now I am trying to understand what is ...
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1answer
98 views

Cohomology of affine plane with double origin

How to calculate cohomology $H^1(X,O_X)$,$H^2(X,O_X)$ $H^1(X,O_X^*)$ of affine plane with double origin $X=\mathbb{A}^2\cup_{\mathbb{A}^2-\{0\}}\mathbb{A}^2$? To use Cech cohomology, I cannnot find a ...
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1answer
50 views

Is there a characterization of contractible hypersurfaces in $\mathbb{C}^2$.

Let $V$ be an irreducible, algebraic hypersurface in $\mathbb{C}^2$ which is contractible as a topological space. I would like to know the algebraic characterization of such objects. For example, ...
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1answer
53 views

direct and inverse images of sheaves and some canonical morphisms

Consider a continuous map $f\colon X\to Y$ between topological spaces. Let $\mathcal F$ be a sheaf on $X$ and $\mathcal G$ a sheaf on $Y$ (let's say of abelian groups). There exists canonical ...
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37 views

What are some elementary books which discuss projective lines on surfaces with examples?

I have the books: W. H. Blythe, On models of cubic surfaces (1905) and A. Henderson, The twenty-seven lines upon the cubic surface, and a couple more modern algebraic geometry books including I. R. ...
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34 views

What are some applications of parametrization of curves and surfaces?

I know that we can find all elements of a quadratic field with norm 1 by rational parametrization of conics, it can be used to show that some Diophantine equations are not so easy to solve, and that ...
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1answer
32 views

Show that $\mathbf{I}(\mathbf{V}(x^n, y^m)) = \langle x, y \rangle$.

Show that $\mathbf{I}(\mathbf{V}(x^n, y^m)) = \langle x, y \rangle$. Where $\mathbf{I}$ is the ideal, and $\mathbf{V}$ is the affine variety. I'm not sure how to even begin on this one. I know ...
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64 views

immersions and finite morphisms

I have the following question: Let $X \subset \mathbb A^n$ be an affine variety. Prove that the immersion $i\colon X \hookrightarrow \mathbb A^n$ is a finite morphism. I know that the ...
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2answers
54 views

Restriction maps in an integral scheme are injective

Suppose $X$ is an integral scheme. I would like to show that the restriction maps $res_{U,V} : O_X(U) \rightarrow O_X(V)$ is an inclusion so long as $V$ is not empty. I was wondering if someone could ...
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1answer
66 views

Why do intersection of two quadratic forms implies elliptic curve?

Let $k$ be a field and $S=k[T_0,T_1,T_2,T_3]$ and $f,g\in S_2$ two relatively prime quadratic forms. How can I show that the intersection $X\subset \mathbb P_k^3$ of second degree surfaces $V_+(f)$ ...
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2answers
105 views

Why Zariski topology is not Hausdorff

I am reading the book about Algebraic geometry. I am confused about the following two things the book mentioned: Zariski topology is 1. different from the topology studied in real and complex ...
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1answer
30 views

Sparse & Dense Polynomials

I've been reading up on Bernstein's theorem for an algebraic geometry assignment and I've come across the terms "dense" and "sparse" in relation to the polynomials. However, I have been unable to find ...
2
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1answer
29 views

Showing a set is an affine variety

I am trying to work through Hartshorne's book and while working through one of the exercises, I need to show the following: Let $k$ be an algebraically closed field. Let $Y \subseteq A^3$ be the set ...
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1answer
35 views

suggestion for a book on algebraic curves

I don't know if i'm asking in the wrong place, but I'm studying the book Algebraic curves of Fulton, and having some problems understanding the final chapters (6,7 and 8). I've found a good help in ...