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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Degree of filtered vector bundle

Suppose I have the sheaf $\mathscr{M}$ defined by $$0\to \mathscr{M}\to \mathscr{O}_{\mathbf{P}^r}^{r+1}\to \mathscr{O}_{\mathbf{P}^r}(1)\to 0 $$ that is, $\mathscr{M}\simeq \Omega_{\mathbf{P}^r}^1(1)...
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1answer
38 views

A question on very ample line bundle on Abelian Varities

I have a problem with some consideration that Mumford does about very ample line bundles in the prove of Riemann-Roch theorem. Namely, he says that if we consider a very ample line bundle $L=O(D)$ on ...
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0answers
32 views

Using Division Algorithm on Polynomials in Finite Field

From Ideals, Varieties, and Algorithms - Cox, Little, O'Shea. Chapter 1, Section 4. Ideals, Exercise 13 (b). Show that every $f \in \mathbb{F}_{2}[x,y]$ can be written as $f = A(x^2-x) + B(y^2-y)...
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70 views

What is the geometrical interpretation of Cartier Divisors?

Definition: Let $(s, \mathcal{L})$ be a pair where $s$ is a rational section of the line bundle $\mathcal{L}$. The Cartier divisor is defined as this pair $(s, \mathcal{L})$. My question: What is ...
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60 views

Metric transformation

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
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1answer
58 views

Categorical Quotients and Group Actions on Varities

So I am given that Let $G = Z/dZ$ where d ≥ 1. Let w be a generator for G and let G act on $A^ {n+1}$ via $w(x_{0}, . . . , x_{n})$ = $(wx_{0}, . . . , wx_{n})$. How can I Show that the ...
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1answer
67 views

On the definition of a reductive group.

Wikipedia defines a reductive group $G$ as an algebraic group with trivial unipotent radical. The radical is the connected component of identity in the maximal normal solvable subgroup of $G$. The ...
2
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1answer
65 views

Covering by open subfunctors and epimorphisms of sheaves.

I am trying to learn about the functor of points approach to algebraic geometry. Given the category of locally ringed spaces $GSp$ (geometric spaces) we have a functor $$\mathcal{G}: GSp \to Set^{...
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0answers
21 views

Is $\overline {G^\circ\cdot p}$ a toric variety?

Consider the algebraic torus $(\mathbb C^*)^n$. Let $G$ be a subgroup of $(\mathbb C^*)^n$ that is also a reductive group. Let $G^\circ$ be the connected component of $G$ containing the identity ...
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1answer
68 views

Coherent sheaves with no cohomology over a hypersurface

Let $X_d \subset \mathbb{P}^{n+1}$ be a smooth hypersurface of degree $d$. How one can describe all coherent sheaves on $X_d$ with no cohomology i.e. $$ H^i(X_d, F) \cong 0, $$ for all $i \in \mathbb{...
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23 views

Implicitization Problem on Graphs?

I learnt the implicitization problem for varieties in introduction course on Algebraic Geometry. I am trying to understand how to formulate a similar implicitization problem on graphs where the ...
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36 views

Why is a smooth curve irreducible? [closed]

As the title suggests, why is a smooth curve irreducible?
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75 views
+300

Does this proof (Lie-Kolchin) suffer from a loss of injectivity?

In the following proof (after "But there is a more elementary proof"), I was confused on something. Apparently we can assume without loss of generality that $V = V_{\chi}$. In this case, here is ...
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0answers
45 views

Fibres of a morphism

Let $f:X\rightarrow S$ a proper morphism, and $s\in S$ a point. If $S$ is locally Noetherian, then what are the properties of the fibre scheme $X_s$ over the Spec of the residue field at $s$? Is this ...
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2answers
52 views

Residue field of the integral closure of a local ring in its field of fractions

When considering the discrete valuation rings contained in the rational functions field $R(F)$ of an irreducible plane projective curve $F \in \mathbb{P}^2(K)$ ($K$ algebraically closed), one can find ...
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43 views

Tensor product of invertible sheaf is still invertible sheaf

In general, the tensor product of two sheaves is the sheafication of the tensor product of two presheaves. So how can we see the tensor product of locally free sheaf with rank 1 is locally free with ...
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48 views

To solve multivariate polynomial equations

For a system of multivariate polynomial equations like this: $$ \left( {\begin{array}{*{20}c} {\frac{{124}} {3}} & { - 24} & {\frac{{ - 68}} {3}} & {\frac{{68}} {3}} \\ {32} & {...
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53 views

Natural isomorphism $\Gamma_\ast(\mathscr{F})^{\tilde{}} \to \mathscr{F}$

This question concerns Proposition 5.15, II, Hartshorne, which states that the natural map $\beta \colon \Gamma_\ast (\mathscr{F})^{\tilde{}} \to \mathscr{F}$ is an isomorphism of $\mathcal{O}_X$ - ...
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27 views

Lifting points via étale morphism of adic spaces

This question was suggested to me during the reading of Huber's book about Etale Cohomology of Adic Spaces. I formulate this question here in the context of adic spaces, but I think, since a morphism ...
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117 views

Points with power distances

It's possible for seven points to be at integral distances. I'm dallying with powered triangles, though, so I'm looking for point sets where all distances are powers of a fixed $x$ value. For example,...
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1answer
51 views

Behaviour of an étale morphism under Galois action on points.

Consider the following situation. Let $k$ be a characteristic $0$ field, and consider an étale morphism of $k$ schemes $f:X\rightarrow Y$. Moreover, let $K$ and $L$ be two extension fields of $k$ such ...
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1answer
68 views

Splitting a short exact sequence of complexes of vector spaces

It's well-known that any complex of vector spaces is isomorphic to a direct sum of two types of indecomposable complexes (a one-dimensional space concentrated in one degree, or two one dimensional ...
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1answer
84 views

Cohomologies with line bundle vs. coherent coefficients

I recently learned in a lecture that the derived category of a smooth variety is generated/spanned by (complexes of) locally free sheaves. (Unfortunately I haven't been able to find a more precise ...
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1answer
41 views

Algebraic geometry look on space-filling curves

Can space-filling curves be somehow described in terms of algebraic geometry? It appears to me that they shouldn't, but I'm not sure. Does anyone know of interesting papers on space-filling curves?
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39 views

“External” Lawvere-Tierney Topologies?

Suppose I have a map $j : \text{Sub}(1) \to \text{Sub}(1)$ from subterminal objects of a topos to themselves which satisfies analogous axioms to those of a Lawvere-Tierney topology, namely $j(1) = 1$, ...
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1answer
30 views

Open vs Closed Immersions of Locally Ringed Spaces

I'm reading Qing Liu's book at the moment and I'm trying to figure out why open immersions of locally ringed spaces are required to be isomorphisms on stalks, but closed immersions are only required ...
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20 views

Does there exist an algebraic group $G$ such that all semisimple elements with infinite order lie in $G^0$?

Let $G$ be a linear algebraic group over an algebraically closed field $k$. If $G$ is connected, and $1_G \neq s \in G$ is semisimple, then $s$ lies in a nontrivial torus: this is Proposition 6.4.5 ...
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27 views

Exercise about Ext functor and 'tri-module'

While trying to understand the Hochschild-Kostant-Rosenberg theorem, I learned that $Ext_{R \otimes R}^1(R, R) = Der_K(R)$, where $R$ is an regular affine (commutative) $\mathbb{C}$-algebra. I am ...
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0answers
71 views

Pushforward of sheaves on the blowup of $\mathbb A^2$ to $\mathbb P^1$

In http://arxiv.org/abs/1210.2564 Example 4.12 it is written that for $Y$ the blowup of $\mathbb A^2$ at the origin (i.e. $Y \cong \mathrm{Tot} \, \mathcal O_{\mathbb P^1} (-1)$), and $π \colon Y \to \...
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1answer
49 views

Genus of the curve $y^2=x^3+x^2$ via Riemann-Hurwitz

How should one apply the Riemann-Hurwirz formula to calculate the genus of the curve $y^2=x^3+x^2$? If I project the x-coordinate to the projective line, I get $2g-2=2(2(0)-2)+3$ since there are 3 ...
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1answer
55 views

Relation between projective equivalence and linear equivalence of divisors

For the whole question I'll be working in $\mathbb{P}^n_{\mathbb{C}}$ and assume that everything is smooth. We say that two sets $U,V\subseteq \mathbb{P}^n$ are projectively equivalent if there ...
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43 views

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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66 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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0answers
50 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
52 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
41 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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54 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
50 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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31 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
56 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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0answers
47 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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65 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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37 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
98 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
61 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
38 views

Tensor product of the stricture 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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34 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
35 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
65 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 (...