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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Isomorphism of polynomial rings [duplicate]

I am trying to do exercise 3.6.F in Ravil Vakil's algebraic geometry notes : http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf We fix a field $k$. It comes down (or so I think) to proving ...
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21 views

Algebraic methods to compute the cohomology ring of the complex topology of a variety?

Suppose $V$ is an affine (resp. projective) subvariety of the affine (resp. projective) space $\mathbb A_\mathbb C^n$ (resp. $\mathbb P_\mathbb C^n$) with vanishing ideal $I\subseteq\mathbb ...
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1answer
50 views

Projective bundles

I am studying about projective bundles now. And I have the following doubts. 1) If we have an exact sequence of vector bundles over a scheme $X$, $0\longrightarrow E'\longrightarrow E\longrightarrow ...
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Generalization of Singular locus and non-free locus to an algebra

Let $R$ be a commutative noetherian local ring with maximal ideal $\mathfrak{m}$ and $ \Lambda $ be a noetherian $ R $-algebra. Recall that: (1) The singular locus of $R$, denoted by $\mathsf{Sing} ...
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1answer
102 views

A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
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21 views

commutativity taking the complement and taking fibers

Let $\mathcal M \rightarrow S$ be a projective irreducible scheme over the spectrum of a DVR and $U\subset \mathcal M$ an open subscheme surjective on $S$. Is it true for both points (generic and ...
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24 views

How many $g_3^1$ does a smooth $(3,3)$ type curve on $\mathbf{P^1}\times\mathbf{P^1}$ has?

Suppose $C$ is a smooth curve of type $(3,3)$ type curve on $\mathbf{P^1}\times\mathbf{P^1}$. Does the two projections provide all the $g_3^1$s for $C$?
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1answer
27 views

Non hyperelliptic curves of genus 5 form a family of dimension 12

Suppose $C$ is a complete intersection of three quadrics in $\mathbb{P}^4$, how to count naively the dimension of its parameter spaces? One needs $|O_{\mathbb{P^4}}(2)|=14$ parameters to describe one ...
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1answer
65 views

Compactifying $\mathcal{O}_{\mathbb{P}^1} (-2)$

I have the total space of $\mathcal{O}_{\mathbb{P}^1} (-2)$ and I see that a "standard" way to compactify is to add the trivial line bundle, $\mathcal{O}_{\mathbb{P}^1}$, and then projectivize. That ...
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24 views

Elementary transformation of vector bundles - equivalent definition

The background is as follows: Let $S$ be a locally noetherian scheme, $E$ a vector bundle over $S$ of rank $N+1$. Let $X=\mathbb{P}(E)$ be the projective bundle and $\pi:X\longrightarrow S$ be the ...
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49 views

How do I calculate the area the Wigner Seitz cells cover in a square?

It's my first time here, so I appologise in advance if I break any rules through this post. So I have a Cartesian Lattice spanning across the Euclidean plane and a unit square. The lattice points ...
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1answer
38 views

Why $(n \times Id )_* O_{A\times A^\vee} = \oplus_{\tau \in A^\vee(S)} (Id\times \tau \circ \pi^\vee)^*P$

Consider an abelian scheme $\pi: A\rightarrow S$, with dual abelian scheme $\pi^\vee: A^\vee\rightarrow S$. The paper I am reading proved a lemma saying that $[n]_* O_A = \oplus_{\mu \in ...
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42 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 $$D(f)=\bigcup_{i=1}^n D(f)\cap Z_i. $$ ...
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72 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. ...
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40 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 ...
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2answers
75 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 ...
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51 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} ...
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1answer
32 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 $U$ ...
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38 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 ...
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1answer
60 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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54 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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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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34 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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32 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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32 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
33 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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23 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 ...
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1answer
37 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) ...
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1answer
49 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 ...
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74 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
78 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
46 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
89 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
204 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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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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39 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
36 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 ...
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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 ...
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37 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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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 ...
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0answers
46 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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1answer
58 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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20 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
54 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
69 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
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 ...
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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
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 ...