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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Question on the second part of the definition of sheafification

I do not understand part $(2)$ of Proposition-Definition 1.2 on page $64$ of Hartshorne's Algebraic Geometry: The original texts are: 'Given a presheaf $F$ ... $F^{+}(U)$ is the set of functions $s$ ...
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71 views

How to show $Hom(V,V)\rightarrow Hom(V_x,V_x)$ is injective, V being semi-stable

Let $V$ be a semi-stable vector bundle over a smooth irreducible projective curve of genus $g\geq 2$. Let $x\in X$. How do we show that the canonical map $Hom(V,V)\rightarrow Hom(V_x,V_x)$ which ...
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51 views

Hartshorne Exercise 2.3.6

Let $X$ be an integral scheme. Show that the local ring $\mathcal{O}_{\xi}$ of the generic point $\xi$ of $X$ is a field. Proof Idea: Let $U \subset X$ be an affine open so that $U= Spec \hspace{...
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1answer
53 views

Let $\phi:A\to B$ be a ring homomorphism, $\phi^{*}:Y\to X$ the induced continuous map on $X=\mathrm{Spec}(A), Y=\mathrm{Spec}(B)$.

This is from Atiyah and MacDonald, Exercise 1.21, part iii). We let $Z=\mathrm{Spec}(R)=\{\mathfrak{p}\subset R\mid\mathfrak{p}\mathrm{\,a\,prime \,ideal}\}$ have the Zariski topology, i.e. with ...
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1answer
47 views

Definition of degree of a coherent sheaf

Let $E$ be a coherent sheaf on a scheme $X$. Let $d = \text{dim}X$ be the dimension of $E$. Huybrechts and Lehn define the degree of $E$ to be: $$ \text{deg} E := \alpha_{d-1}(E) - \text{rk}(E)\cdot\...
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64 views

Quadrics intersecting the twisted cubic and a line.

I am trying to understand the determinantal approach on Harris book "Algebraic Geometry: A first course" on proving that the intersection of two quadrics containing the twisted cubic in $\mathbb{P}^3$ ...
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1answer
57 views

What does it mean for a scheme to be proper?

What exactly does it mean for a scheme to be proper? I can't seem to find an actual definition of this anyway despite the term being frequently used.
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37 views

Is any abelian subvariety of an abelian variety closed?

I am wondering because Milne here in Proposition 10.1, page 42, takes any abelian subvariety $B$ of an abelian variety $A$, with $0\neq B\neq A$, then he takes an ample line bundle $\mathcal{L}$ on $A$...
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1answer
61 views

Deriving Formulae for the roots of the quartic and cubic polynomials

I have seen derivations of the general solution for the roots of fourth and third degree polynomials of 1 variable in Dummit & Foote's Abstract Algebra; however, it was by no means simple to me. I ...
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61 views

Chern classes of a double cover

Let $X$ be a compact complex surface and let $D$ be a double cover of $X$. Let $\pi:D\to X$ be the double cover map (a 2:1) map. If $E$ is a vector bundle (rank at least 2) on $X$ with $c_1(E) = A$ ...
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71 views

Why doesn't $R(G)$ contain a torus?

$G$ is a connected linear algebraic group which is not solvable, and $T$ is a maximal torus of $G$, with $\textrm{Dim } T = 1$. $B$ is a Borel subgroup of $G$ containing $T$, and $U$ is the group of ...
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40 views

Fibre of morphism of schemes.

Hartshorne gives the following preamble to the definition of a fibre of a morphism of schemes. Let $f: X \to Y$ be a morphism of schemes, and let $y \in Y$ be a point. Let $k(y)$ be the residue ...
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66 views

Multiplication map between sheaves on $\operatorname{Proj}A$

Let $A$ be a graded ring, $X=\operatorname{Proj}A$ and let $f$ be an element of degree $d>0$. I have come across the phrase "Let $\mu\colon\mathcal{O}_X \to \mathcal{O}_X(d)$ be the map given ...
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2answers
100 views

Prove that $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ is an integral domain [closed]

Let $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ be an analytic algebra. I am trying to prove that $R$ is an integral domain. Basically I know that if $\langle x^2 - yz\rangle$ is a prime ...
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1answer
65 views

The non-existense of the fine moduli scheme of vector bundles. Why?

The reference I am using is this enter link description here. The question is about the moduli space of vector bundles. I am trying to understand why the fine moduli scheme does not exist. Let $C$ a ...
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1answer
58 views

Tensor product of the structure sheaf with the function field

Consider an irreducible scheme $X$ with function field $K(X)$. Then define the presheaf $$U\mapsto \mathscr O_X(U)\otimes_{\mathscr O_X(U)} K(X)$$ for every open set $U\subset X$. Is this presheaf ...
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35 views

Torus-invariant divisor on $\mathbb{CP}^2\#1\mathbb{CP}^2$

I want to confirm that the following result is right: $M=\mathbb{CP}^2\#1\mathbb{CP}^2$. I was told that the torus-invariant divisors on $M$ are $H,H-E,E$, where $E$ and $H$ are the hyperplane ...
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1answer
37 views

Confusion over Grobner Bases, Division algorithm, and ideal memebership.

I'm reading through Justin Smith's Introduction to Algebraic Geometry. Before getting into coordinate rings, he talks about Grobner bases. He's given a division algorithm in which given and ordering ...
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1answer
66 views

3-dimensional measure of a permutahedron

Let $P = \operatorname{conv}\lbrace (s(1), s(2), s(3), s(4)) \mid s \in S_4 \rbrace$ be a permutahedron. Compute the 3-dimensional measure of this polytope. I know that $P$ is three dimensional (...
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80 views

The $2 \times 3$ matrices with rank $\leq 1$ cannot be defined by two polynomial equations

Let $X$ be the space of all ${2 \times 3}$ matrices over $\mathbb{C}$ that have rank at most 1. This is naturally a subspace of $\mathbb{C}^6.$ We can express $X$ using 3 polynomial equations, namely ...
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1answer
84 views

Eisenbud-Harris Exercise II-14, limit scheme isomorphic to triple point and remembers both tangent line, osculating $2$-plane to subscheme

Let $C$ be the subscheme of $\mathbb{A}_K^n$ given by the ideal$$J = (x_2 - x_1^2, x_3 - x_1^3, \ldots).$$A closed point in $C$ is of the form $f(t) = (t, t^2, t^3, \ldots, t^n)$, for $t \in K$; that ...
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24 views

intersection between conic and cubic

Let $ A=V(\alpha)$ a non-singular conic in $\mathbb{P}^2(\mathbb{C})$, $V(\delta_1)$ and $V(\delta_2)$ two cubic such that : $D_1$ and $D_2$ meet $A$ tangentially at six distincts points $P_1,...,P_6$ ...
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83 views

Global sections and divisors

I'm trying to understand the proof of the Theorem at page 163 from Mumford, Abelian Varieties, and I have a question about one step. This is the situation: $X$ is an abelian variety (hence there's ...
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1answer
106 views

Strict transform of blow up

Let $X$ be a smooth projective variety over $\mathbb{C}$. Consider the blow up of $X$ about a closed subvariety $Z$. Let $X'=Bl_Z(X)$. Let $Y$ be a smooth irreducible divisor of $X$ properly ...
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0answers
65 views

Why is the intersection of algebraic subsets of an algebraic variety again an algebraic subset?

This question is inspired by the Wikipedia article on the Zariski topology: https://en.wikipedia.org/wiki/Zariski_topology Since I know next to nothing about algebraic geometry, and no advanced ...
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1answer
45 views

Determining the projective closure of a variety.

Consider the twisted curve in $\Bbb{C}^3$ denoted by the ideal $\langle x^2-y, z-xy\rangle$ I did it by homogenizing the generators: I got $\langle x^2-yw, zw-xy\rangle$. The projective closure ...
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1answer
37 views

Isogenic elliptic curves. Number of points and zeta function

Is there any book or other reference where I can find a complete proof of the following fact? If $E$ and $E'$ are two isogenic curves (over $\mathbb{F}_q$, where $q=p^a$), then for any $n \ge 1$ the ...
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1answer
53 views

Definition of pullback of a Weil divisor on an abelian variety?

We are on an abelian variety, so Cartier divisors, Line bundles and Weil divisors are all equivalent. I would like to see the pullback of a Weil divisor. Is it true that, if $D=\sum n_i E_i$, then the ...
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2answers
52 views

restrictions of rational sections of an invertible sheaf.

Let $(X,\mathscr O_X)$ be an integral scheme with function field $K(X)$ and let $\mathscr L$ be an invertible sheaf on $X$. Moreover suppose that $\{U_i\}$ is an open covering of $X$ such that $\...
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189 views

Is there a universal statement for the construction of global Proj?

Let $X$ be a scheme and $\mathcal{A}$ be a sheaf of $\mathbb{Z}_{\ge 0}$-graded $\mathcal{O}_X$-algebras. From the data above there is a construction which gives the "global Proj" $Proj \mathcal{ A} ...
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20 views

local representation of “logarithmic connection”

Let X be a Riemann compact surface, $D\subset X$ be a finite subset, and (E,$\nabla$) be a logarithmic connection. And let $z$ be a local coordinate at $p\in D$, why $\nabla $ can be written by: $\...
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1answer
53 views

Zariski closure of $T:= \{(t_1,t_2,t_3,t_1t_2t_3^{-1})|t_i\in \mathbb{C}^*\}\subseteq \mathbb{C}^4$?

Let $V= \mathcal{V}(\langle xy-zw\rangle)\subseteq \mathbb{C}^4$ be an affine variety. The set $T:= \{(t_1,t_2,t_3,t_1t_2t_3^{-1})|t_i\in \mathbb{C}^*\}$ is a torus contained in $V$. I am trying to ...
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1answer
63 views

Given a locally ringed space there is a bijection between open and closed sets of $X$ and idempotent elements of $\mathcal{O}_X(X) $

This is a problem from Gortz and it does NOT assume that the underlying space is the spectrum of the ring or anything like that. Now I proved easily that given a clopen set of $X$ there is an ...
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0answers
61 views

Roots of canonical line bundles that are not necessarily square roots

I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are $2^{2g}$ choices of such a root. But what about further roots? ...
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1answer
31 views

The isotropy of the action of $SU(3)$ on $\mathbb CP^2$

Consider the action of $SU(3)$ on the complex projective plane $\mathbb CP^2$. How we can find the isotropy group?
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43 views

A complement of Milnor's book on singularities with exercises/examples

I'm currently reading "Singular Point of Complex Hypersurfaces" by Milnor. This is really a great book, but I did realize I didn't saw any concrete examples, computations etc ... I was wondering if ...
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0answers
47 views

Obtaining a nice map to a curve by using blowups

Let $X$ be a smooth and projective variety over a finite field (separated, finite type, integral). Then after performing a number of blowups I should be able to find a proper surjective map from $X$ ...
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4answers
194 views

Diagonal morphism of regular variety is a regular embedding

Let $X$ be a regular $k-$variety (i.e. all of its local rings are regular) of pure dimension $d$. Then I would like to show that the diagonal morphism $X\rightarrow X\times_k X$ is a regular embedding ...
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0answers
103 views

Is this true that, any algebraic curve has finitely many singularities?

Can we say that any algebraic curve has finitely many singularities?
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51 views

Covering maps of schemes.

A curve $X$ is modular if there is a finite covering $X_0(N)\rightarrow X$. What does covering mean in this context, and for more general morphisms of schemes? Just covering as topological spaces?
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2answers
59 views

Scheme theoretic 'class inclusions'

For lack of a better name*, I call the following two things class inclusions: $$1) \quad\textbf{Magma}\supset \textbf{Semigroup}\supset\textbf{Monoid}\supset \textbf{Group}\supset \textbf{Abelian ...
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1answer
85 views

Cohomology of the Munford line budle on an Abelian variety

Let $X$ be an Abelian variety over a field $k$; $L$ line bundle on $X$. I would like to calculate the cohomology of the Mumford line bundle $\Lambda(L)=m^*L\otimes p_1^*L^{-1}\otimes p_2^*L^{-1}$; ...
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0answers
34 views

Explain Betti diagram of a minimal free resolution for a simplical complex

I am self-learning algebraic geometry and reading the book The Geometry of Syzygies A Second Course in Algebraic Geometry and Commutative Algebra and I want to ...
2
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1answer
58 views

The isotropy of the complex projective plane for the action of $SU(3)$

If we consider the action of the compact real form $SU(3)$ of $SL(3,\mathbb C)$ on the space $\mathbb C^3$. Since the action is transitive, how to find the stabilizer $G_x$? Is it useful to find ...
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0answers
64 views

Riemann-Roch and quartic

I know very little in algebraic geometry, but I want to learn!! So I know the Riemann-Roch theorem as follow: let $$L(D)=\{\text{ meromorphic functions, s.t. }\operatorname{div}(f)\geq D \}$$ and $$...
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1answer
95 views

Hilbert polynomial for a dimension zero projective variety by taking an affine chart

I am looking at exercise 12.21 from Gathmann's notes on algebraic geometry. I am given a homogeneous ideal $$I \unlhd k[x, y, z] $$ with a dimension $0$ projective locus. WLOG, we assume that this ...
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46 views

Exterior Algebra VS Torsion

Let $C$ be an irreducible and reduced rational curve, and $f: \mathbb P^1\rightarrow C$ be the normalization. If $\mathcal F$ is a coherent sheaf of rank $r$ over $C$, then I was wondering if we can ...
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1answer
34 views

To solve large systems of multivariate polynomial equations

Nicolas Courtois et al. proposed the eXtended Linearization(XL) method to solve the systems of multivariate polynomial equations and analyzed the time complexity. Polynomial when the number of (...
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73 views

Vanishing of Cohomology of Affine Schemes — Proof

I was following Ravi Vakil notes FOAG (http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf) and I cannot understand Th.18.2.4, at least not all of it. What I can follow is the first part: ...
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106 views

Understanding an application of Riemann-Roch in an article

I saw the following in an article: Let $C$ be an irreducible smooth projective curve over an algebraically closed field $K$ and let $g$ be its genus. By Riemann-Roch, if N is large enough for every ...