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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Irreducible component of a scheme over a dvr

Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
2
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1answer
51 views

What's wrong with this calculation involving pullbacks of divisors on surfaces?

Beauville, Complex Analytic Surfaces, Proposition I.8(b), reads: Let [S and] $S'$ be a surface, $g : S \to S'$ a generically finite morphism of degree $d$, and $D$ and $D'$ divisors on $S$. Then ...
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0answers
42 views

Is the constant group scheme for $\mathbb{Z}$ affine?

Is the constant group scheme for $\mathbb{Z}$ affine? It is said no in Gille's notes "INTRODUCTION TO REDUCTIVE GROUP SCHEMES OVER RINGS" 3.1, but I don't see why!
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21 views

Three and a half basic questions on the Weil restriction of scalars

I am currently trying to get familiar with the Weil Restriction functor. For a finite field extension $L|K$ it associates a variety over $K$ to every variety $X$ over $L$ as the representing object ...
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30 views

Open subscheme in special fiber

Let $X$ be a projective scheme over $R$ a discrete valuation ring with generic fiber irreducible. Can an open subscheme of $X$ be contained in the special fiber of $X$ ? Or is it true that every open ...
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30 views

Translate a geometric theorem into polynomial equations — Theorem of the orthocenter of a triangle

This is Exercise 13 of Chapter 6 of Ideals, Varieties, and Algorithms by Cox et al. The problem asks to translate the following geometric theorem into polynomials and using Groebner basis to test ...
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0answers
28 views

An exact sequence of Chow groups

Let $X$ be a closed subscheme of $\mathbb{P}^n$, with canonical line bundle $O(1)$,let $V\subset \mathbb{A}^{n+1}$ be the affine cone over $X$. How to show there is an exact sequence ...
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1answer
28 views

Is taking projective closure a functor?

For an affine variety $X\subset \mathbb{A}^n$, we can associate it with $\overline{X}$, which is the closure of $X$ in $\mathbb{P}^n$. Does $\overline{X}$ depend on the choice of embedding ...
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20 views

Disk in analytic topology vs. the spectrum of a Henselian DVR in etale topology

In this informative and concise set of notes on vanishing cycles by Donu Arapura, it is stated that the theory of vanishing cycles ports nicely to the etale world if the role of the disk is replaced ...
2
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1answer
33 views

Attempt at understanding Weierstrass points

I'm reading through Springer - Riemann surfaces and Farkas and Kra - Riemann surfaces and theta functions. I'm attempting to get an understanding of Weierstrass points. I've come up with a (hopefully) ...
4
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1answer
44 views

Smooth Fano Polytopes and Hypersurfaces

This is a rather extended question, so I will try to make it as compact and readable as possible. I am trying to practice with the Macaulay2 software, in particular the polyhedra and ...
2
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0answers
70 views

K-theory of projective space

Is there any way to prove that the twisting sheaves $\mathcal{O}(K)$ generate the algebraic K-theory of projective space without actually using any K-theory machinery (e.g. Bott periodicity)? Like for ...
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1answer
74 views

Tensor product of $\mathscr{O}_X$-modules which results in a presheaf.

Background: Over a locally ringed space $X$, if we define the tensor product of two $\mathscr{O}_X$-modules $\mathscr{F}$ and $\scr{G}$ naively as $U \mapsto \mathscr{F}(U) \otimes \mathscr{G}(U)$, we ...
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2answers
45 views

generalized principal open set

Let $V$ an affine variety. A principal open set is a set of the form $V(f) = V \setminus\{f=0\} $. A well known theorem states that all such sets are affine varieties, and moreover (Shafarevich, p.50) ...
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1answer
85 views

Geometric interpretation of cubic curve?

Lines and conics have clear geometric meanings that are coordinate-free, but cubics seem to rely entirely on cubic equations and coordinate systems. Are there ways to define cubic curves without cubic ...
2
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1answer
53 views

Do analytic functions on open subsets of $\mathbb{C}$ with an analytic square root form a sheaf? [duplicate]

I'm trying to learn algebraic geometry and am trying to think about what kinds of things are presheafs but not sheafs. One exercise I had was to show that bounded holomorphic functions on open ...
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1answer
149 views
+100

Hypercube and Hyperspheres

Let $n,k\in\mathbb{N}$. In this problem, the geometry of $\mathbb{R}^n$ is the usual Euclidean geometry. The lattice hypercube $ Q(n,k)$ is defined to be the set $ \{1,2,...,k\}^n ...
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0answers
28 views

What is a good lex order to compute the Groebner basis of this ideal?

This comes from chapter 6 of Ideals, Varieties and Algorithms by Cox et al. The equations are from a planar robot with three joints and one prismatic joint. See the following picture: Given a ...
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0answers
38 views

When is the Zariski closure of subset connected [closed]

Let $R$ be a commutative ring and $\{P_i\}_{i\in I}$ be an arbitrary subset of $Spec(R)$ such that $\dfrac{R}{\bigcap_\limits{i\in I}{P_i}}$ is an indecomposable ring, how can we show that the ...
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1answer
35 views

Function Field of Variety and Scheme

Let $V\subseteq \mathbb{A}^n_k$ be a closed irreducible algebraic set ("affine variety") over a closed field $k$. Construct the topological space $X$ consisting of all closed irreducible subsets of ...
2
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1answer
48 views

Why does isomorphism follow from the natural bijection of Hom sets

If X and Y are varieties, and Y is affine, there is a natural bijective mapping of sets $$\operatorname{Hom}(X,Y)\xrightarrow{\sim}\operatorname{Hom}(A(Y),\mathscr O(X))$$ where the left are ...
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36 views

infinite-order elements of $Out(\widehat{F_2})$

Let $\widehat{F_2}$ be the pro-$\ell$ completion of the free group of rank 2, where $\ell$ is some prime. Every outer automorphism of $F_2$ induces an outer automorphism of $\widehat{F_2}$, hence an ...
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0answers
51 views

“Affine Variety” versus “Variety”

In Mumford's Red Book, a distinction seems to be made between the terms affine variety and variety. The term variety seems to be defined by something called the Hausdorff. I was wondering if anyone ...
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0answers
44 views

Calabi-Yau Toric Varieties

This is a rather naive question, but, from what I understand, we begin with a some reflexive polytope $P$. From the basic theory of toric varieties, we can construct a toric variety corresponding to ...
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0answers
53 views

Simple example of the use of sheaves

What would be (one of) the simplest example of a mathematical result which is solved using the concept of sheaves?
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17 views

Hasse-Weil zeta function of projective hypersurfaces

Assume $f$ is a homogeneous integer polynomial in $n\geq 3$ variables such that the hypersurface $f=0$ is irreducible over $\mathbb{Q}$ (but not necessarily over $\overline{\mathbb{Q}}$ so for example ...
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1answer
51 views

Identifying two points on an algebraic curve

Given a smooth algebraic curve $C$, say projective over an algebraically closed field $k$, is it always possible to identify two distinct closed points $x, y$ on $C$ to produce a curve with a single ...
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2answers
68 views

Triviality of a vector bundle is an open condition

Is the following statement true (and if not, are there additional assumptions that make it true?) A vector bundle on a variety which is trivial if restricted to a closed subvariety is trivial on ...
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0answers
34 views

Difference between $K$-rational points and $K$-valued points

I am not entirely sure if there is a difference between $K$-rational points on a scheme $X$ over $k$ and $K$-valued points on $X$. Both seem to refer to a $k$-morphism Spec $K \to X$ but the ...
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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?
1
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1answer
41 views

A question about current and Dirac measure

$0$ can be seen as a divisor of $\mathbb{C}$, and the current $[0]$ is defined as $[0](\varphi)=\varphi(0)$. Why is this reasonable?
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1answer
32 views

Determining the fourth vertex of a parallelogram knowing that its the point of intersection of two circles

This question was part of the exercises in one of the courses i'm taking. The answer was already provided. The first circle was assumed to have as its center, vector $v_1$, while its radius was the ...
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 ...
2
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1answer
63 views

How does one find the Zariski closure of a set?

I've started to learn algebraic geometry this week (so I do not have much knowledge in the subjet) and, after reading the definition of the Zariski closure $V(I(S))$ of a set $S$, I've tried to do the ...
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 ...
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 ...
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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 ...
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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."
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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 ...
2
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1answer
113 views

What is the equation describing a three dimensional, 14 point Star?

I need to model a 14 point star. This is a three dimensional surface where there is a point at each of the eight corners of a cube and each of the six sides. The object is uniform (i.e. planar ...
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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 ...
4
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1answer
83 views

Is the cuspidal curve $\mathcal{M}$ is a coarse moduli space for lines in $\mathbb{C}^2$?

As the question suggests, is the cuspidal curve $\mathcal{M}$ a coarse moduli space for lines in $\mathbb{C}^2$? I'm inclined to believe the answer is no, but all attempts at proving it so far have ...
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0answers
55 views

What is the degree of the pull back of a line bundle?

Let $X=\mathbb{P}^2$, and let $(y_1,y_2,y_3)$ be homogeneous coordinates on $X$. Consider a map $\phi:\mathbb{P}^1\longrightarrow X$, given by $\phi(x_1,x_2)=(x_1^2,x_1x_2,x_2^2)$, where $(x_1,x_2)$ ...
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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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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 ...
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 ...
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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 ...
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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 ...
3
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1answer
83 views

Affine varieties over finite fields

I read in this paper (http://www.math.iitb.ac.in/~srg/preprints/Chandigarh.pdf) that the following set is an affine variety: $V_f=\{(t_0,...,t_N)\in \mathbb{F}_p^{N+1} : f(t_0,...,t_N)=0 \}$ where ...