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. ...

learn more… | top users | synonyms (1)

0
votes
0answers
17 views

What is $\overline{D(f)}$?

Let $A$ be a ring, $f\in A$. If $A$ is Noetherian, $\text{Spec}(A)$ has finitely many irreducible components, let us call them $\{Z_i\}_{i=1}^n$. So we write $$D(f)=\bigcup_{i=1}^n D(f)\cap Z_i. $$ ...
3
votes
0answers
26 views

What is the group structure on the ring of power series around a point that makes it “the completion of an elliptic curve” along that point?

I've been struggling to understand the explicit details of the completion of an elliptic curve about the origin, and am desperately confused by the explicit details of the resulting group operation. ...
1
vote
0answers
24 views

Gluing Schemes, Closed Subschemes

Let $X$ be a scheme and $Y$ a closed subset. Take a covering of open subsets $U_i$ of $X$ which are affine. Say $U_i\simeq \text{spec } A_i$, choose $\mathfrak{a}_i$ to be the largest ideal with ...
2
votes
2answers
54 views

Exercise 6.5.F in Ravi Vakil's notes: Showing conic $x^2 + y^2=z^2$ in $\mathbb{P}_k^2$ is isomorphic to $\mathbb{P}_k^1$

I have been stuck on Exercise 6.5.F in Ravi Vakil's notes for a little while now, and I would greatly appreciate any hints/comments/solutions! Let $k$ be a field that is not of characteristic $2$. I ...
0
votes
0answers
34 views

Zeta function, how to solve a finite geomatry summation.

I wanted to solve the zeta function for an undifend period "$d$". So for every $d\ge2$. $$\zeta(-s)= \frac{1}{(d^{s+1}-1)}\sum_{m=1}^{\infty} \frac{1}{2^{m+1}}\sum^{m}_{j=1} ...
1
vote
1answer
27 views

Does every non trivial variety in $\mathbb{R}^n$ have empty interior?

By this question, we know that a non-trivial affine variety in $\mathbb{C}^n$ has empty interior. But the argument uses the (strong) fact that a holomorphic function vanishing in a non empty set $U$ ...
0
votes
0answers
35 views

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
votes
1answer
57 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 ...
1
vote
0answers
49 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!
2
votes
0answers
23 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 ...
0
votes
0answers
32 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 ...
2
votes
0answers
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 ...
3
votes
0answers
30 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 ...
1
vote
1answer
30 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 ...
1
vote
0answers
21 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
votes
1answer
35 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
votes
1answer
47 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
votes
0answers
72 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 ...
7
votes
1answer
76 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 ...
1
vote
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) ...
2
votes
1answer
88 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
votes
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 ...
14
votes
1answer
172 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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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
votes
1answer
49 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 ...
3
votes
0answers
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 ...
0
votes
0answers
52 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 ...
1
vote
0answers
45 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 ...
1
vote
1answer
56 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?
0
votes
0answers
18 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 ...
4
votes
1answer
53 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 ...
0
votes
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 ...
0
votes
0answers
35 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 ...
0
votes
1answer
23 views

Sampling points uniformly from arbitrary region. [closed]

What are some techniques for sampling points uniformly from a semi-algebraic set?
1
vote
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?
0
votes
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
votes
1answer
31 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
votes
1answer
64 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
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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
votes
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
votes
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 ...
-1
votes
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?
1
vote
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
votes
1answer
86 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 ...
1
vote
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 $(x_1,x_2)$ ...