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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45 views

Background for reading Atiyah's first paper on the twisted cubic

What should an undergraduate know before being able to read Michael Atiyah's A Note on the Tangents of a Twisted Cubic ? Most of the words in the paper look foreign to me, but I'm very intrigued by ...
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1answer
24 views

smooth affine algebraic curves and their subschemes

I am reading a lot about curves at the moment and I am a little confused: Let $X= Spec K[X]$ denote a smooth affine algebraic curve. Then, according to some sources, the ring $K[X]$ is a Dedekind ...
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1answer
35 views

how to distance circles drawn on another circles

I need to do some calculations in order to do this drawing (sorry for the quick sketch): I need to define a set of variables and do simple calculations as much as possible in order to come up with ...
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1answer
52 views

Zariski topology on Affine spaces, Name of Functor

I've been studying the Zariski topology in my free time. So I found this functor between Polynomial Algebras and Affine Spaces. First, we have this $T$ such that for any affine space ...
5
votes
1answer
58 views

Functor of points definition of a space modeled on a site

I'm trying to find a definition of a space modeled on a site which is: (i) plausible and natural in the context of general sites (ii) subsumes common examples. Let $(C,J)$ be a grothendieck site and ...
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1answer
40 views

Algebraic families of vector spaces that are pairwise dependent

Let $T \subset Gr(n,2n)$ be an algebraic family of complex $n$-dimensional vector subspaces in $\mathbb{C}^{2n}$, $n > 1$, and denote the vector space corresponding to a point $t$ of $Gr(n,2n)$ by ...
2
votes
1answer
37 views

Suppose X is a closed subscheme of Y, with Y locally Noetherian. Is there a locally free resolution of $i_* O_X$?

Let $I : X \to Y$ be a closed subscheme of a locally Noetherian scheme. I am secretly trying to show that sheaf exts of $i_* O_X$ to coherent sheaves on Y are coherent (in order to find a dualizing ...
4
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0answers
61 views

Tropicalization of a line in the projective plane P^2

Lets assume that our field $K$ is the Puiseux series. I have been working with tropicalization from the book "Introduction to tropical geometry" link : ...
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0answers
54 views

Separable morphism and smooth fibers

Let $f:X \to Y$ be a separable, dominant morphism of finite type between noetherian $k$-schemes for $k$ algebraically closed. Does it mean that For a closed point $x \in X$, $f^{-1}(f(x))$ is smooth ...
2
votes
2answers
46 views

Example of a dominating map

Unfortunately the book that i am reading (Algebraic curves by Fulton) has no examples, so i am trying to find an example of a dominating map that would be helpful for the understanding of the ...
2
votes
1answer
29 views

The canonical base point for Weil algebras

Kock defines, after (16.2), the canonical base point of a small object $\operatorname{Spec}_R(W)$ to be $$\mathbf 1\overset{\operatorname{Spec}_R (\pi)}{\longrightarrow}\operatorname{Spec}_RW$$where ...
4
votes
2answers
90 views

Fiber of morphism homeomorphic to $f^{-1}(y)$

I want to solve exercise 3.10 (a) of Hartshorne's book, chapter II, which asks to prove the following: Let $f\colon X\to Y$ be a morphism of schemes and let $y\in Y$, then $X_y=X\times_Y ...
3
votes
3answers
59 views

Why are the irreducible components $T$-stable?

I'm having trouble with part of a proof (7.1.5) in Springer's Linear Algebraic Groups. Let $r: G \rightarrow \textrm{GL}(V)$ be a rational representation of a linear algebraic group $G$, $B$ a Borel ...
3
votes
1answer
37 views

Function field, finite extension, isomorphism implies isomorphism?

Let $A$ be a function field in $1$ variable over $\mathbb{C}$, and let $B$ be a finite extension of $A$ of degree $[B : A]$. If $B \cong \mathbb{C}(x)$ over $\mathbb{C}$, then does it necessarily ...
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0answers
17 views

Image absolute Frobenius by $G_x \longrightarrow \mathrm{Gal}(\kappa(x)/\kappa(y))$

Let $X$ be a quasi-projective scheme over the finite field $\mathbb{F}_q$ and $G$ a finite group acting on $X$. Let $Y = X/G$ be the quotient scheme and $\pi : X \longrightarrow Y$. Let $x \in X$ be a ...
0
votes
0answers
29 views

Buchberger algorithm and ideals

I'm working on Groebner bases using the book Ideals, Varieties and Algorithms. I'm interested in this problem : Let $\mathbb{Q}[x,y,z]$ with the graded lexicographic order with $x>y>z$. For ...
1
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0answers
18 views

Action of Torus on Grassmanian - a Highest Weight Description, or otherwise intrinsic description

What is an intrinsic description of the action of the Torus on the Grassmanian = $GL(n)/P$, where $P$ is a certain parabolic subgroup? The explicit description in terms of the Plücker embedding I ...
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0answers
15 views

Does the trivial character always show up as a weight?

Let $G$ be a linear algebraic group, $T$ a subtorus of $G$ of dimension $\geq 1$. Let $\mathfrak g$ be the Lie algebra of $G$. Then the Ad operator $$\textrm{Ad } : G \rightarrow ...
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0answers
20 views

Functoriality of $\text{Proj }A$

This is a remark from Ulrich Gortz's Algebraic Geometry - Functoriality of $\text{Proj }A$ - Let $A=\bigoplus_{d\ge0}A_d$ be a graded ring. We can "thin out" $A$ and "change $A_0$" without ...
4
votes
0answers
86 views

Understanding the topology of $y^2=(x-1)(x-2)(x-3)(x-4)$

Andreas Gathmann's lecture notes on algebraic geometry start by considering the curve $C_n=\{(x,y): y^2 = (x-1)(x-2)...(x-2n)\} \subset \mathbb{C}^2$. He claims that the topology of this curve is the ...
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0answers
18 views

Is affine GIT quotient necessarily an open map?

Let $k$ be a field, $X=$Spec$A$ be an affine scheme with A a f.g. $k$-algebra. $G=$Spec$R$ is a linearly reductive group acting rationally on A. (i.e. every element of $A$ is contained in a finite ...
3
votes
1answer
51 views

Why do we need the infinite field hypothesis in this cohomology calculation?

I've just finished my very first calculation with sheaf cohomology. It's exercise III.2.1(a) in Hartshorne, and it says Let $X = \mathbb{A}_K^1$ be the affine line over an infinite field $K$. Let ...
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1answer
98 views

Line bundle trivial on fibers then isomorphic to the pullback of a line bundle

$\require{AMScd}$ I'm currently reading Milne's notes about Abelian varieties. On page 26 he proves the following theorem: Let $V$ and $T$ be varieties over $k$ with $V$ complete, and let ...
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votes
2answers
27 views

Dimension of irreducible component of reduced ring

Let $X= SpecA$ denote the spectrum of a reduced ring $A$. Is there any way to tell the dimension of an irreducible component $Y$ of this variety? Each irreducible component corresponds to a minimal ...
3
votes
2answers
82 views

Sheaf cohomology with support

Let $X$ be a topological space and $\mathcal F$ is a sheaf of abelian groups on it. Let $Y$ be a closed subspace of $X$. Let $\mathscr{H}^0_Y \mathcal F$ be the subsheaf of $\mathcal F$ with supports ...
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votes
2answers
67 views

$\mathbb{P}_{\mathbb{C}}^3$ is not isomorphic to $S^2 \times S^4$

I have been trying to solve this exercise given by my prof. The hint is to show that every $2$-form $w$ on $S^2 \times S^4$ is s.t. $w \wedge w = 0$, while this is not true in case of ...
3
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0answers
44 views

If f/g is symmetric (resp homogeneous), must f and g be as well?

Suppose we have two polynomials $f$, $g$ in $k[X_1, ..., X_n]$ over some field $k$, and they have no factor in common. Suppose that $f/g$ is symmetric. Must than $f$ and $g$ also both be symmetric? ...
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1answer
41 views

do formal group laws induce group structures on schemes (as opposed to formal schemes)

Let $R$ be a ring and $f \in R[[x]]$ a commutative formal group law over $R$, meaning $f(f(x, y), z)=f(x, f(y, z))$, $\ f(x, y)=f(y, x)$ and $f(x, y)=x+y + \text{higher order terms}$. Let ...
2
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0answers
13 views

Some clarifications regarding the definition of the Hilbert-Mumford weight

I am reading about the Hilbert-Mumford criterion and I am stuck at something that is stated as "obvious" in every text that I can find. A bit of help would be much appreciated. So, let $X$ be a ...
4
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0answers
60 views

Clifford, $p$-forms and spinors

I'm trying to understand the paper by Atiyah, Hitchin and Singer called: ''Self-duality in four dimensional Riemannian geometry", available here. I'm stuck at the point where it explains how the ...
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0answers
31 views

product of divisors on the square of a curve

Let $C$ be a smooth projective curve over a field $k$ and let $D$ be a divisor on $C$. I have seen people considering a zero-cycle on the surface $C \times C$ which they denote by $D \times D$. If ...
0
votes
1answer
26 views

Why is $T/S$ isomorphic to $k^{\ast}$? (Remark 7.1.4 in Springer Linear Algebraic Groups)

I had a quick question about quotients of varieties. I am still not very good at them. Let $T$ be a torus, $\alpha$ a nontrivial character of $T$, and $S = (\textrm{Ker } \alpha)^0$. Since $T$ is ...
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0answers
36 views

How can I define an isomorphism here?

Let $V$ be a variety in $A^n$(affine n-space with algebracally closed field k base) , $I=I(V) \subset k[X_1,X_2,...,X_n], P \in V $, and let $J$ be an ideal of $k[X_1,X_2,...,X_n]$ that contains ...
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vote
1answer
54 views

Understanding problem 2.6 in Hartshornes algebraic geometry book .

Can anybody help me in understanding the hint given in the problem $2.6$ in Chapter $1$ of Hartshorne's Algebraic Geometry book ? I cannot see why $A(Y_i) $ can be identified with degree zero ...
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0answers
32 views

An algebraic set is called defined over $k$ if its ideal can be generated by polynomials in $k[x]$. [duplicate]

I find this definition in Silverman's book, The Arithmetic of Elliptic Curves. An algebraic set (in $A^n(\bar{K})$) is called defined over $K$ if its ideal can be generated by polynomials in ...
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0answers
30 views

What's the dimension of the space formed by taking a union of projective lines between two spaces?

I've read that the subspace formed by taking the union of all projective lines that join a point of $\mathbb{P}^{l}$ with a point of $\mathbb{P}^{m}$, where $\mathbb{P}^{l}$ and $\mathbb{P}^{m}$ are ...
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votes
0answers
22 views

Can Hecke Operators be defined on more general spaces of elliptic curves?

Classically, the Hecke Operators act as endomorphisms of $\omega^k_{\mathcal{M}_{ell}(\mathbb{C})}$, defined by noting that there is a distinguished class of covering maps $\widetilde{E}\to E$ given ...
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1answer
67 views

Intuition about formal brances of a curve at a point

Consider an algebraic surface $X$ and a curve $Y\subset X$. Here $X$ is a $K$-scheme integral of finite type of dimension $2$ and $Y$ is a closed subscheme of dimension $1$. Fix a closed point ...
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0answers
43 views

How to describe a polynomial relation on $\mathbb{P}(\bigwedge^k V)$, and if the Zariski topology is canonical

I am working with the space $\mathbb{P}(\bigwedge^k V)$, where $V$ is some $n$ dimensional vector space over some field K. In here I want to define a variety, ie a solution to a set of polynomials. ...
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votes
1answer
41 views

Fiber of morphism induced by map on stalks

Given a morphism of schemes $f\colon X\to Y$ and a point $x\in X$, the map on the stalks induces a morphism $\operatorname{Spec}\mathcal{O}_{X,x} \to \operatorname{Spec}\mathcal{O}_{Y,f(x)} $. Is it ...
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0answers
44 views

Dimension of the stalk at the point in the closure of an open subscheme

Let $U$ be an open subscheme of $X$ a $\mathbb C$-scheme locally of finite type. I know that $U$ is of dimension $n$ and I have a point $x$ in the closure of $U$. What can be said of the Krull ...
3
votes
1answer
78 views

Exercise $2$ from chapter $5$ of Eisenbud's Geometry of Syzygies book

I am trying to solve exercise $2$ from chapter $5$ of Eisenbud's The Geometry of Syzygies book.The problem is as follows: Let $X$ be the union of two disjoint lines in $\mathbb P^3$, or a conic ...
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votes
2answers
48 views

Is the ideal of a variety the annihilator of a subspace of the symmetric algebra?

Let $V$ be a vector space over an algebraically closed field $K$. Let $\mathrm{Sym}(V^*)=\mathrm{Sym}(V)^*$ be the symmetric algebra on $V$, i.e. if we give a basis $e_1,...,e_n$ of $V$ and let ...
0
votes
1answer
60 views

Derivative of a section of a vector bundle

Let $X$ be a complex algebraic variety and let $E \to X$ be a vector bundle over $X$, with sheaf of sections $\mathcal{E}$. If $s$ is a local section of $\mathcal{E}$, what is the derivative $ds$ ...
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votes
1answer
30 views

Variety of homogenous polynomials which factor as the product of linear forms

Let $V$ be the complex vector space of all homogenous polynomials in three variables of degree $d$ and $\mathbf{P}V$ the corresponding projective space. Let $X$ be the subset of all those homogenous ...
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1answer
35 views

Regularity and Short Exact Sequence

Suppose $ 0 \to M_1 \to M_2 \to M_3 \to 0$ is a short exact sequence of finitely generated graded $k[x_0,...,x_r]$-modules. Then show that $\mathrm{reg}(M_1) ...
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0answers
35 views

Trying to Compute Regularity and degree

Definition: For a finite subset $X \subset \mathbb P^r$,the Hilbert function $H_X(d)$ is constant for large $d$ and its value is the number of points in X,usually called the degree of $X$. Let ...
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votes
1answer
28 views

Torus action and multigrading.

Let $G$ be an algebraic group and $T$ the maximal torus. Suppose that $T$ acts on $G$. Do we have a multigrading on $\mathbb{C}[G]$? How to define the multigrading corresponding to the $T$-action? ...
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vote
1answer
38 views

Problem in understanding the proof of lemma $7.2.5$ in Liu's book

Let's analyze the proof of the following lemma: Lemma: Let $X$ be an integral, Noetherian scheme and let $f\in K(X)^\ast$, then for all but finitely many points $x\in X$ of codimension $1$ we have ...
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votes
1answer
41 views

A basis for forms of degree $d$ (Fulton, 2.35)

I am trying to solve this exercise from Fulton's book: (2.35)(c) Let $L_1, L_2, \dots,$ and $M_1, M_2, \dots$ be sequences of nonzero linear forms in $k[X,Y]$ and assume no $L_i = \lambda M_j$ for ...