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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1answer
22 views

Sampling points uniformly from arbitrary region. [closed]

What are some techniques for sampling points uniformly from a semi-algebraic set?
4
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2answers
44 views

Third point of intersection is also a point of inflection?

Let $C \subset \mathbb{P}_2$ be a nonsingular cubic. If $L$ is a line through two distinct points of inflection on $C$, how do I show that the third point of intersection is also a point of ...
3
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1answer
30 views

Representation-theoretical reasons for positivity of product of two Schubert polynomials?

In the Wikipedia article on Schubert polynomials there is a claim that there are representation-theoretical reasons for the product of two Schubert polynomials to have nonnegative coefficients when ...
0
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1answer
38 views

injective morphism between line bundles on curves

Let $X$ be a smooth projective, irreducible, curve, $\mathcal{L}$ be an invertible sheaf on $X$ and $\mathcal{L}' \subset \mathcal{L}$, an invertible subsheaf. Is $\deg(\mathcal{L}') \le ...
3
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2answers
69 views
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0answers
35 views

Linearizable reductive group action.

Let $k$ be a 0 characteristic field, and $G$ a reductive group in $GA_2(k)$ (the group of automorphisms of k[x,y] as k-algebra). How is it possible to deduce that $G$ is conjugated to a subgroup of ...
2
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1answer
53 views

Reference on Malcev completion

I need a reference for learning Malcev completion, its associated group scheme, and Lie algebra. Thanks!
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0answers
28 views

Self-intersection number of fibered surface

Let $R$ be a complete discrete valuation ring with residue field $k$ (can assume algebraically closed), $f:X \to \mathrm{Spec}(R)$ a flat, proper family of projective curves (i.e., $f$ is also ...
2
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1answer
66 views

Colimits and tensor product ground rings

Is it true that $\varinjlim (M \otimes_{A_i} N) = M \otimes_A N$ where $A = \varinjlim A_i$ and $M$ and $N$ are $A$-modules? Take maps $f : A_j \rightarrow A_k$ and $m : M \otimes_{A_j} N \rightarrow ...
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0answers
15 views

finite subgroups of O(3,1) [closed]

What are the finite subgroups of $O(3,1)$ or $O(p,q)$ more generally? Apparently "this question body does not meet our quality standards."
2
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0answers
42 views

Weil: Fibre Spaces in Algebraic Geometry

I have spent a decent amount of time searching for the notes for Weil's Fibre Spaces in Algebraic Geometry, written by A. Wallace, both in print and online. Does anyone have a file of it they'd be ...
4
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0answers
49 views

How to prove these claims about ideal sheaves?

The following claims come from the proof of Proposition 3.10 (Page 66) of D.Huybrechts' Fourier-Mukai Transforms in Algebraic Geometry. Since I couldn't find these results in Hartshorne's Algebraic ...
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0answers
39 views

Question about Poincare series

Let $R=\mathbb Q[x,y]_{(x,y)}$ and $I=(x^{10},x^8y,xy^4,y^5)$. Then how can we calculate the Poincare series of $I$ by Macaulay 2?
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0answers
25 views

Properties of a scheme with a closed morphism to a normal crossing divisor

All schemes here are noetherian, separated and of finite type over $\mathbb{C}$. Let $D=V_+(xy)\subset \mathbb{P}^2=Proj(\mathbb{C}[x,y,z])$ be the normal crossing divisor comming from the coordiante ...
1
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1answer
30 views

Complete intersection curve

I have two very basic questions/clarifications. Let $X=\mathbb{P}^n_k$, and let $Y$ be a subvariety of $\mathbb{P}^n_k$ of dimension $m$. Then we say that $V$ is a complete intersection variety if ...
3
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2answers
43 views

Continuity of Galois representations from cohomology

One of the most standard way to construct Galois representations is the geometric way: one starts from a variety $X$ defined over $\bf Q$ say; the Galois group acts on ${\overline X}:= X \times {\rm ...
2
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1answer
39 views

For which varieties is the natural map from the Chow ring to integral cohomology an injection?

For a smooth projective complex variety $X$ over $\mathbb{C}$, there is a natural map from its Chow ring $\mathbb{A}^*(X)$ into even integral cohomology $H^{2*}(X)$ of its (often implicitly ...
3
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0answers
107 views

Remark 4.23.4 in Hartshorne.

Remark 4.23.4 in Hartshorne references a paper by Elkies that explains that$$\mathfrak{B} = \{p \text{ prime}: X_{(p)} \text{ is nonsingular over }k_{(p)}, \text{ and }X_{(p)}\text{ has Hasse ...
1
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1answer
49 views

Distinguished points of a cone

Sorry, as this is a rather trivial question that I am misunderstanding, but I do not understand how the distinguished point is defined. We define it as a homomorphism from some semigroup $S_{\sigma}$ ...
0
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2answers
21 views

If $W$ admits an injection of $k$-algebras in its coordinate ring, then $W$ is an unirational variety

I'm studying algebraic geometry from "Introduction to algebraic geometry" by Hassett, and I did not understand a step in his proof of the following result (page 52): "If $W$ is an affine variety ...
0
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1answer
31 views

Equation of the hyperplane that passes through points on the different axes

We work over $\mathbb{R}^N$. I have a set of points, each of which is on a different axis. For instance, when $N=3$ the set is given by $S=\{ (p_1,0,0);(0,p_2,0);(0,0,p_3) \}$, where $p_1$, $p_2$, and ...
0
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0answers
27 views

$R^kp_{2,*} (p_1^* V\otimes P) =0$ for $k\neq g$ and $V$ is $\pi_*$ acyclic on abelian schemes?

Let $\pi: A\rightarrow S$ be an abelian scheme of relative dimension $g$, and $A^\vee$ its dual, and $P$ the Poincare bundle. We have the projections $p_1,p_2$ from $A\times A^\vee$ to $A$ and ...
4
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1answer
49 views

Hartshorne's proof of the exact sequence $\mathbb{Z} \to \operatorname{Cl} X \to \operatorname{Cl} U \to 0$

Hartshorne, Algebraic Geometry, Proposition II.6.5 reads (in part): Let $X$ satisfy (*), let $Z$ be a proper closed subset of $X$, and let $U = X \setminus Z$. Then: [...] (c) if $Z$ is ...
3
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1answer
49 views

Stalks of the sheaf of total quotient rings

Let $X$ be a scheme, for each $U$ open in $X$, let $S(U)$ be the set consisting of elements of $O_X(U)$ whose image in $O_{X,p}$ is a non-zerodivisor for every $p\in U$. In particular, if $U = ...
4
votes
1answer
321 views

Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space ...
3
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0answers
32 views

A non-singular quotient of $\mathbb{A}^n$ by a cyclic group is isomorphic to $\mathbb{A}^n$

Let $G$ be a cyclic group acting linearly on $X := \mathbb{A}^n$. If we assume that the quotient $Y:=X/G$ is non-singular, does it follow that $Y \simeq \mathbb{A}^n$? If so, is it necessary to assume ...
0
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0answers
48 views

Are Prevarieties irreducible?

In Goertz-Wedhorn, a prevariety is defined to be a connected space with functions that locally is an affine variety (were an affine variety is a space with functions that is isomorphic to the space ...
1
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0answers
18 views

Polynomials as Locally Isotrivial Covers

Let $k=\mathbb{A}^1$ be algebraically closed of arbitrary characteristic. I am interested in understanding when a polynomial $f:\mathbb{A}^n\to\mathbb{A}^1$ defines a locally isotrivial family over ...
1
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1answer
27 views

A question on very ample line bundle and closed immersion

Let $X$ be a projective scheme, $i:X \hookrightarrow \mathbb{P}^n$ a closed immersion, $\mathcal{L}:= i^*\mathcal{O}_{\mathbb{P}^n}(1)$ a very ample line bundle. Let $j:{\mathbb{P}^n} \hookrightarrow ...
2
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0answers
40 views

How to show two varieties are NOT birationally equivalent?

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Some backgroud It comes from a problem showing that $Q=V(x^2+y^2-z^2-1)$, a hyperboloid, and $W=V(x+1)$, a plane, are ...
2
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1answer
24 views

What is the ideal sheaf of a translated subvariety of an abelian variety?

Let $A$ be an abelian variety (over an algebraically closed field $k$) and $X\subset A$ a nonsingular subvariety with ideal sheaf $\mathscr I_X\subset \mathscr O_A$. Let $\tau_a:A\to A$ be the ...
1
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2answers
36 views

Closure of subset of affine plane

Inspired by this question, I wonder if one can prove the following Let $ k $ be an algebraically closed field. Is the closure of $ \{(x,y):x^{2}+y^{2}=1,x\ne 0\} $ in the affine plane over $ k $ ...
1
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1answer
28 views

Codimension is preserved under base change for given conditions

I'm working on exercise II.6.1 in Hartshorne and I'm stuck on the following step. I would appreciate some help. Let $ X $ be a Noetherian integral scheme. Let $ Z $ be a closed subscheme of ...
2
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0answers
50 views

On why $k(X)^{G}$ is a finitely generated field extension

In a book I was reading, from the assumptions that we have a linear algebraic group $G$ acting on an irreducible (affine) variety $X$, the author writes that $k(X)^{G}$ is a finitely generated field ...
4
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1answer
63 views

Additivity of the first Chern class

I have a highly elementary question: is the first Chern class additive? More specifically, given a short exact sequence of coherent sheaves on a nonsingular curve $X$ $$ 0 \to \mathscr F'\to \mathscr ...
3
votes
1answer
188 views

Computing the trajectory of an orbiting body so that it collides with another orbiting body

I am creating a 2D game in which two space ships, orbiting around a planet under the influence of gravity, fire projectiles at each other, which are also under the influence of gravity. I'm creating ...
2
votes
2answers
61 views

Ideal of polynomials vanishing on $\{(x,y): x^2+y^2=1, x \neq 0 \}$

I'm reading the book "Introduction to algebraic geometry" by Hassett, and in Chapter 3, after introducing the concept of the ideal of polynomials vanishing on a set $S$, the author gives some ...
5
votes
1answer
85 views

What's wrong in my thinking about Bézout's theorem?

First, I know that every hypersurface of degree $d$ defined in $\mathbb{CP}^n$ is diffeomorphic. By using this fact, I wanted to calculate the Euler characteristic of hypersurface of degree $d$. To ...
3
votes
1answer
49 views

Are real algebraic points dense in a real affine variety?

Let $V\subset \mathbb R^n $ be the zero-locus of finitely many polynomials with rational coefficients. Is it true that the set of points in $V$ whose coordinates are algebraic numbers is dense in the ...
6
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1answer
31 views

composing with a non-flat morphism

Let $X,Y,Z$ be integral schemes and $f:X\to Y$, $g:Y\to Z$ be morphisms such that $g$ is flat and $f$ is not flat. Does this imply that the composition $g\circ f:X\to Z$ is necessarily non-flat? ...
0
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0answers
28 views

Lines on a specific cubic surface

Consider the cubic surface given in affine coordinates by the equation $x^2+y^2=g_3(z)$ ($g$ is a polynomial of degree 3 s.t. the cubic is smooth). Is it possible to write down explicitly the ...
0
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0answers
17 views

Algorithmic question about algebraic varieties and affinely independence

Let $f \in \mathbb{R}[x_1,\ldots,x_d]$ be a polynomial in $d$ variables. Then we can write $f$ as $$ f = \sum_{i=0}^m f_i x_1^i $$ where $f_i \in \mathbb{R}[x_2, \ldots, x_d]$. We may assume $m \neq ...
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1answer
46 views

best introductory simple readable undergraduate book for algebraic geometry [duplicate]

I like to study algebraic geometry and what is the best introductory simple readable undergraduate book for it?
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0answers
19 views

Something about closed sets in $\mathbb{P}^n\times\mathbb{P}^m$

The closed sets in $X:=\mathbb{P}^n\times\mathbb{P}^m$ are generated by sets of the form $X\setminus(U\times V)$ where $ U\subseteq\mathbb{P}^n $and $V⊆\mathbb{P}^m$ are open. I was mentioned about ...
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0answers
36 views

Definitions of $\mathcal{O}(n)$ and sheaves associated to a module

In Vakil's Foundations of Algebraic Geometry (see p. 385) he defines the sheaf $\mathcal{O}(m)$ on $\mathbb{P}^n$ to be equal to $\mathcal{O}$ on affine sets $U_i = $Spec $k[x_{0/i}, x_{1/i}, \dots, ...
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0answers
29 views

Is there any book or paper about the rational intersection points between two hyperelliptic curves?

I have two hyperelliptic curves and i want to know the rational intersection points between them
0
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0answers
23 views

Morphisms between Prevarieties

I'm trying to understand the attached proof from Mumford's Red Book. In particular, I am confused by the statement that "$g•f$ is at least a section of $O_X$ on the sets $f^{-1}(V \cap {V_i})$."I see ...
8
votes
1answer
73 views

Zeta function, $\mathbb{F}_5[T, \sqrt{T(T-1)(T+1)}]$

Let $A = \mathbb{F}_5[T, \sqrt{T(T-1)(T+1)}]$. My question is, what is the easiest way to see that$$\zeta_A(s) = {{1 + 2 \cdot 5^{-s} + 5^{1 - 2s}}\over{1 - 5^{1 - s}}}?$$Much thanks in advance. ...
3
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1answer
125 views

Proof verification of a weak version of Bezout's Theorem

I'd like to make sure here that my reasoning seems sound. I am working from Kirwan's book on algebraic curves. I was not totally happy with her proof of this theorem, so I wanted to see if I could ...
0
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0answers
44 views

When are direct products exact in the category of quasi-coherent sheaves?

I would like to know if there is a description (or at least some sufficient condition known) of a (Noetherian) schemes $X$ such that the category $\mathrm{QCoh}_X$ does have exact direct products. I ...