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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2
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40 views

When is $f^! \mathcal{O}_Y$ a line bundle?

Let $f: X \to Y$ be a finite, surjective morphism of reduced, separated schemes of finite typer over some field of characteristic zero. The sheaf $\mathcal{H}om_Y(f_* \mathcal{O}_X, \mathcal{O}_Y)$ is ...
0
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0answers
27 views

open sets in affine space are not affine varieties - easy proof

In a previous post, I have defined a generalized principal open set: $$V_{f_1,\dots,f_n}=\mathbb{C}^m\setminus \{f_1 = \dots = f_n =0\}$$ where $f_1,\dots,f_n$ are polynomials in $m$ variables. This ...
-1
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0answers
43 views

A question regarding Grothendieck , topos and (adelic??) points

I am having a look at this conference: https://www.youtube.com/watch?v=yNgvvNx_P9w I am particularly interested in getting your feedback on 1:14:30 and the seconds thereafter. Could anyone explain ...
2
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0answers
38 views

Geometric intuition for flatness

The definition of flatness, as given for example in Hartshorne, is essentially algebraic and not very intuitive geometrically (for me at least). What are good geometric ways to think about ...
1
vote
1answer
39 views

Localization of a coherent module is coherent

I would like to prove that the coherence of a sheaf on a scheme is affine-local. For this, it is necessary to prove that for a coherent $A$-module $M$, its localization $M_{f}$ at $f\in A$ is a ...
5
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1answer
68 views

Why would we a priori expect $V(I)$ to satisfy axioms to define the closed sets for a topology on $\text{Proj}(S)$?

The topological space $\text{Proj}(S)$ has the underlying set$$\text{Proj}(S) = \{\mathfrak{p} \text{ a homogeneous prime such that }S_+ \not\subseteq \mathfrak{p}\},$$and the closed sets are the loci ...
2
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0answers
31 views

General Fiber in Positive Characteristic

It is well-known that a complex polynomial, considered as a function $f:\mathbb{C}^n\to\mathbb{C}$, is a fiber bundle over a cofinite set of "atypical values" which include the singular values of $f$. ...
3
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0answers
26 views

Etale fundamental group action on set-theoretic fiber

Let $f: Y \to X$ be a finite etale cover of schemes. Fix a geometric point $x \in X$. I would like $\pi_1(X,x)^{et}$, the etale fundamental group, to act on the set-theoretic fiber of $x$. This set is ...
0
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1answer
27 views

Properties of resolution of singularities

Let $X$ be a complex algebraic varieties and $\pi:X' \to X$ be a resolution of singularities of $X$. Let $Y$ be a smooth (irreducible) subvariety of $X$. Is $\pi^{-1}(Y)$ smooth and irreducible? What ...
0
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0answers
19 views

Linear spaces on quadrics

Given a smooth quadric hypersurface $Q\subseteq\mathbb{P}^r$, many properties (dimension, irreducibility etc.) of the Fano varieties $F_{k}(Q)$ of $Q$ (which is the set of $k$ dimensional linear ...
4
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1answer
44 views

Can a bidegree $(3,4)$ curve be embedded in plane?

Suppose $C$ is a curve on $\mathbf{P}^1\times\mathbf{P}^1$ of bidegree $(3,4)$, why such a curve cannot be embedded as a curve in $\mathbf{P}^2$?
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0answers
17 views

On the definition of the Period Domain for K3 surfaces

By definition, the Period Domain of K3 surfaces is $$\Omega = \{[x]\in\Bbb{P}(\Lambda\otimes\Bbb{C}): \ x^2=0, \ x\overline{x}>0\} $$ where $\Lambda$ is the K3 lattice. The two relations $x^2=0$ ...
1
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1answer
37 views

$\mathbb{C}$-points on a $\mathbb{Z}$-scheme

Let $\mathcal{X}$ be a "nice" scheme over $\mathbb{Z}$. We could assume $\mathcal{X}$ regular and $f: \mathcal{X} \rightarrow \mathbb{Z}$ flat and projective, but feel free to change these ...
5
votes
1answer
96 views

When $\cos(\theta) = 1/8$ it's easy to show $\theta$ is an irrational angle. Is it algebraic?

Along the lines of my lines of my previous question about irrational angles "$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?", I was working on ...
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0answers
28 views

When does a morphism from variety over another variety descend to a morphism from the base variety?

Suppose I have a morphism of varieties $f:W\times \mathbb{P}^1 \to \mathbb{P}^1$, where $W$ is a curve. I have an \'{e}tale cover $p:W \to V$. Suppose that the map $f(w_0,\cdot)=f(w_1,\cdot)$ for all ...
1
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0answers
20 views

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 ...
1
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0answers
42 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 ...
1
vote
1answer
53 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 ...
-1
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0answers
34 views

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} ...
7
votes
1answer
110 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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0answers
31 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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0answers
29 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 $C$?
1
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1answer
33 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 ...
3
votes
1answer
75 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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0answers
25 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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0answers
72 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 ...
1
vote
1answer
39 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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0answers
44 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 $$D(f)=\bigcup_{i=1}^n D(f)\cap Z_i. $$ ...
5
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0answers
85 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
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1answer
48 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
77 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
54 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
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 $U$ ...
0
votes
0answers
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 ...
2
votes
1answer
61 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
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!
3
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0answers
30 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
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0answers
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 ...
2
votes
0answers
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 ...
3
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0answers
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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0answers
26 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
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) ...
4
votes
1answer
51 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
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 ...
7
votes
1answer
79 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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vote
2answers
53 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
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
votes
1answer
54 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 ...
15
votes
1answer
239 views

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 ...