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Questions tagged [algebraic-geometry]

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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68 views

Testing if two finite sets of points differ only by rotation (in polynomial time)?

Imagine we have two size $m$ sets of points $X=\{x^i\}_{i=1..m}, Y=\{y^i\}_{i=1..m} \subset \mathbb{R}^n$ and we need to answer the question if they differ only by rotation: if there exists othogonal $...
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66 views

Why are sheaves called sheaves?

By "why" I mean: what were the stated intentions of the namers? More generally, how did the theme of agricultural terminology in algebraic geometry come about?
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47 views

Can a complex manifold that is not a Calabi-Yau manifold be homeomorphic to a Calabi-Yau manifold?

This is a kind of a follow up to this question, which actually already had an answer here, in which it is asserted that Hodge numbers in general are not topological invariants. Could it be so extreme ...
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1answer
23 views

Double dual of $\mu$-semistable sheaf

Is it true that if $F$ is a $\mu$-semistable sheaf, then its double dual is also $\mu$-semistable? We know $c_1(F)=c_1(F^{**})$. Given a sub sheaf $E$ of $F^{**}$, if we can associate to it some ...
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25 views

More Pisot and Salem numbers

The ten smallest non-quadratic Pisot numbers are given at the link. The smallest is the plastic constant $\rho$ with a value of 1.32471795. All values up to 1.57367896 are known. Does someone have ...
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56 views

Exercise 2.6 in Hartshorne's book.

Exercise 2.6: If $Y$ is a projective variety with homogeneous coordinate ring $S(Y)$, show that $\dim S(Y)=\dim Y+1$. I don't know where is wrong about the following argument. My idea: copy the ...
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1answer
58 views

Are Hodge numbers topological invariants for manifolds that admit a Kähler structure?

I know that all fibers in a analytic fibration (proper, holomorpic) are homeomorphic, and if the fibers are Kählerian manifolds, then they have equal Hodge numbers. Could it happen however that a ...
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87 views

How do we know that the only polynomials $f$ satisfying $f(\lambda x_0,\ldots,\lambda x_n) = f( x_0,\ldots,x_n) $ are the constant polynomials?

In algebraic geometry one proves that the affine coordinate ring of $\mathbb{P}^n$ is trivial by using that the only polynomials $f$ satisfying $f(\lambda x_0,\ldots,\lambda x_n) = f(x_0,\ldots,x_n) $ ...
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1answer
26 views

Linking the cohomology of a coherent sheaf on a curve with the cohomology of its restrictions to irreducible components

$\newcommand{\H}{\operatorname{H}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}}\newcommand{\O}{\mathcal{O}}\newcommand{\I}{\mathcal{I}}$ Let $X$ be a projective scheme of dimension one over ...
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1answer
37 views

Confusion about embeddings of algebraic varieties

I tried to learn a bit about embeddings of algebraic varieties into other algebraic varieties. Depending on the textbook/online notes I consider, there are the notions of closed immersion and open ...
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33 views

Another proof for impossibility of covering $\mathbb{R}^{n}$ with a set of varieties of cardinality less than $2^{\aleph_0}$

Let $\mathbb{R}^{n}$ be the affine $n$-space. Let $\{V_{\alpha} \}_{\alpha \in \Gamma}$ be a set of real varieteis such that $\mathbb{R}^{n} \subseteq {\bigcup V_{\alpha}}.$ Prove that $|\Gamma| \geq ...
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26 views

Geometric role of Zariski's main theorem in structure theory of smooth, unramified, and étale morphisms

In this MO comment, Brian Conrad stresses that Zariski's main theorem is the engine facilitating the structure theory of smooth, unramified, and étale morphisms. In the theory of smooth manifolds, ...
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1answer
47 views

Connectedness of complex sphere

Let $X_{n}$ be a set $$ X_{n} = \{(x_{1}, \dots, x_{n})\in \mathbb{C}^{n}\,:\, x_{1}^{2}+ \cdots + x_{n}^{2} = 1\}. $$ For $n\geq 2$. Then $X_{n}$ is connected. In the case of $\mathbb{R}$, it is ...
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1answer
36 views

Adjunction Formula for Sheaves

Let $f: X \to Y$ be a morphism of ringed spaces, $\mathcal{G}$ a $\mathcal{O}_Y$-module. I have a question about an argument used to construct a adjunct morphism $f^{\#}: \mathcal{G} \to f_*(f^*(\...
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1answer
37 views

Conjugacy class that is not closed

I'm reading Springer's Linear algebraic group and I stuck with one exercise. Let $k$ be a field of characteristic 2 and let $G = \mathrm{SL}_{2}$. Then it says that conjugacy class of the matrix $$ ...
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26 views

Genus of Fibration

Consider a fibration $f : S \to B$ of an algebraic surface $S$, (i.e. a $2$-dimensional, proper $k$-scheme) over a curve $B$ (i.e. $1$-dim "). My question is quite banal, but till now I don't found ...
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24 views

Projective space is characterized by its Hilbert polynomial

Is it true that if a projective variety over an algebraically closed field has the same Hilbert polynomial (respect to a fixed very ample line bundle) as a projective space,then it is a projective ...
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1answer
34 views

The ring of formal series in a variable $R=k[[x]]$ is a local ring, where the ideal maximal $m$ is the ideal of all series with term independent zero

The ring of formal series in a variable $R=k[[x]]$ is a local ring, where the ideal maximal $m$ is the ideal of all series with term independent zero I have already noticed this Maximal ideals of the ...
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29 views

Moduli space of stable curves of genus $g$ does not admit a universal family

I am trying to understand why the (coarse) moduli space $\overline{M}_g$ of stable curves of genus $g$ does not admit a universal family. I am following the proof in p.267 of this book. A key step is ...
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Constructing a group scheme associated to a coherent module.

For a commutative ring $A$ and an $A$-module $M$, $\mathbb{V}(M)$ is defined to be $\text{Spec}(S^*M)$, where $S^*M = \bigoplus_k S^kM$ is the symmetric algebra. Thus $\mathbb{V}(M)$ is an affine ...
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1answer
37 views

Torsion free sheaf on $\Bbb P^2$

Let $F$ be a coherent torsion-free sheaf on $\Bbb P^2$ and $L \subset \Bbb P^2$ be a line. Assume that there is an isomorphism $f : F_{|L} \to \mathcal O_L^r$ for some $r \in \Bbb N$. Questions : 1)...
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1answer
32 views

Is $Y=V(\cos(t),\sin(t),t)\subset k^3$ a variety?

Is $Y=V(\cos(t),\sin(t),t)\subset k^3$ a variety? I think this is not a variety but I do not know how to prove this rigorously, could someone help me please? How does it look on the $k^3$ plane? ...
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30 views

Quadratic forms with common real zeros

Is it possible to find three quadratic forms $f_1,f_2,f_3$ in $K[x,y,z]$ where $K$ is a real number field such that $f_1,f_2$ and $f_3$ have exactly 2 common real zeros $f_1^2+f_2^2+f_3^2$ can be ...
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2answers
59 views

calculate the position of a body moving along a path represented by given function.

Suppose that we have a body that will move over a curve (for example a parabolic curve). The equation of that curve is : $$ y+k=(x+h)^2 $$ Where (h,k) are the (x,y) of the vertex. Also suppose that ...
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1answer
56 views

Yoneda Lemma in Vakil’s FoAG 1.3.Y

I am trying to do the exercise that reads as follows, Suppose you have two objects $A$ and $A'$ in a Category $D$, and morphisms $i_C:Mor(C, A) → Mor(C,A′)$ that commute with the maps $Mor(C,A) → ...
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73 views

Fixed points of an endomorphism of a ring

Let $R$ be a commutative $k$-algebra, where $k$ is a field of characteristic zero. Let $f$ be a $k$-algebra endomorphism of $R$. ($f$ is not assumed to be either injective nor surjective). Are ...
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1answer
71 views

Smooth point implies infinitely many rational points

I'm reading this article on strong approximation, and am coming across the following example of the hypersurface $X$ defined by $3x^3 + 4x^3 + 5z^3 = 0$. The claim is that since any point on $X$ ...
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25 views

Inverse limit and rational points

Let $X \subset \mathbb A_{\mathbb Z}^d$ be an affine scheme of finite type over $\mathbb Z$. Let $I$ be the ideal of $\mathbb Z[t_1, ... , t_d]$ corresponding to $X$, generated by $f_1, ... , f_r$. ...
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24 views

Properties of intersection of subalgebras

Let $R$ be a commutative $k$-algebra, which is an integral domain, where $k$ is a field of characteristic zero. Let $R_i$, $i \in \mathbb{N}$, be $k$-subalgebras of $R$. Denote $A:=\cap_{i \in \mathbb{...
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1answer
38 views

Example of a ring with unique prime ideal of a given height

What is an example of a commutative ring $R$ (with 1) such that dim $R \geq 2$ and there is a unique prime ideal of a given height. Such a ring is of course local and if we assume $R$ is finite ...
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1answer
24 views

Geometric way to see that a morphism of affine varieties is not an isomorphism?

Consider the morphism of affine varieties $\mathbb{A}^1 \rightarrow \mathbb{V}(y^2 - x^3) \subset \mathbb{A}^2$ defined by $t \mapsto (t^2, t^3).$ Clearly the morphism is a bijection and it also seems ...
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55 views

Is the surface topologically unique? [on hold]

I have been doing some geometric modeling and came across this geometric object, I was wondering if it was topologically unique or it belongs to a particular geometric class (Platonic Solids, ...
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1answer
43 views

Hilbert-Chow morphism $Hilb^2(C)\to Sym^2(C)$ for $C$ a singular curve

I would like to know if there is anything known about the Hilbert-Chow morphism for the case of a singular curve. Due to Fogarty, for any number $n$ and any smooth projective variety $X$, there is a ...
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1answer
35 views

Are morphisms of algebraic varieties determined by their underlying functions?

Let $K$ denote an algebraically closed field. Define that an algebraic variety over $K$ is a ringed space that can be covered by open sets, each of which is isomorphic to an affine algebraic variety, ...
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1answer
74 views

Schubert class in the Grassmannian G(3,6)

How to compute the Schubert class $\sigma$$^2$$_2$$_1$ in the Grassmannian G(3,6)? I remember the result is $\sigma$$_3$$_3$ + 2$\sigma$$_3$$_2$$_1$ + $\sigma$$_2$$_2$$_2$.
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73 views

It is the set $V(y-\sin(x))\subset k^2$ a variety?

It is the set $V(y-\sin(x))\subset k^2$ a variety? I know that $V(y-\sin(x))=\{...,-4\pi,-3\pi,-2\pi,-\pi,0,\pi,2\pi,3\pi,4\pi,...\}$. Could it be a set of points a variety? My intuition tells me ...
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1answer
21 views

Show that if $I\subset R$ is an ideal then $I\subset h^{-1}(IS)$. Give an example in the that inclusion is proper

Let $h: R\to S$ be a ring homomorphism. Let $P\subset R$ be a prime ideal. Denote by $PS$ to the set $$PS=\{s\in S: s=\sum_{i \text{ finite}}h(r_i)s_i, r_i\in R, s_i\in S\}$$ Show that if $I\subset R$...
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33 views

Connected levels and polynomials submersions

Is it true that a polynomial submersion $ p: \mathbb{R}^2 \to \mathbb{R}$ of degree $n$ has at most $n$ connected components on each level?
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2answers
27 views

Give an example to show that in general $h(P)$ is not an ideal of $S$.

Let $h: R\to S$ be a ring homomorphism. Let $P\subset R$ be a prime ideal. Give an example to show that in general $h(P)$ is not an ideal of $S$ The first thing I think is to take $R=\mathbb{Z}$ and $...
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1answer
44 views

Equivalent Valuations

Def: A valuation on a ring $A$ (commutative with identity) is a map $\lvert \cdot \rvert : A \to \Gamma \cup \{0\}$ such that for all $a,b\in A$, $\lvert a+b \rvert \leq \max\{\lvert a \rvert, \lvert ...
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1answer
26 views

Show that $PS$ is the smallest ideal of $S$ that contains $h(P)$

Let $h: R\to S$ be a ring homomorphism. Let $P\subset R$ be a prime ideal. Denote by $PS$ to the set $$PS=\{s\in S: s=\sum_{i \text{ finite}}h(r_i)s_i, r_i\in R, s_i\in S\}$$ Show that $PS$ is the ...
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2answers
41 views

Show that if $X$ is an irreducible variety, and if $U,V\subset X$ are open not empty, then its intersection $U\cap V$ is not empty

Show that if $X$ is an irreducible variety, and if $U,V\subset X$ are open not empty, then its intersection $U\cap V$ is not empty I have reasoned in the following way: Reasoning by the absurd ...
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1answer
26 views

Is $M$ an $A$-module with finite presentation?

Given an $A$-module $M$, then we have an $\mathcal O_{\operatorname{Spec}A}$-modules $M^\sim$. If $M^\sim$ is locally of finite presentation, is $M$ an $A$-module with finite presentation?
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1answer
56 views

Regular global sections of invertible sheaves

$\newcommand{\L}{\mathcal{L}}$ $\newcommand{\ox}{\mathcal{O}_X}$ Let $X$ be a projective scheme of dimension one over a field $k$ and let $\L$ be an invertible sheaf on $X$. What are sufficient ...
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1answer
65 views

Is the set $K$ finite or infinite?

Say you have $\Phi(x)=2^{sA_n}$, for some fixed parameter $s\in \Bbb R$, $n \in \Bbb Z^+ $ with $A_n=1/\log(p_n(x)) $, where $p_n(x) $ is a polynomial with degree $n$. How many forms $p_n(x)$ yield ...
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18 views

A descending chain of algebras $R_i \subseteq k[x,y]$ of Krull dimensions $2$, such that $\cap R_i$ is of Krull dimension $1$.

Let $k$ be a field of characteristic zero, and let $R_i$ be a descending chain of $k$-subalgebras of $R_0:=k[x,y]$. Assume that the Krull dimension of each $R_i$ is $2$. I am looking for an example ...
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19 views

Sweeping a G-orbit by a different group H

Suppose H and K are algebraic subgroups of a linear algebraic group G. Suppose that G acts on a smooth algebraic variety X. Given $x\in X$, we can consider the map \begin{equation} H\times K \...
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54 views

Is a subalgebra of $\mathbb{C}[x_1,\ldots,x_n]$ of Krull dimension one, isomorphic to $\mathbb{C}[t]$?

Let $R$ be a $k$-subalgebra of $\mathbb{C}[x_1,\ldots,x_n]$, $n \geq 1$. Assume that $R$ is of Krull dimension $1$. Is $R$ isomorphic to a polynomial ring in one variable? More generally, ...
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0answers
40 views

On purity of structure sheaf of a closed subscheme

I have started reading the book "GEOMETRY OF MODULI OF SHEAVES" by Daniel Huybrechts and Manfred Lehn. I have come across the following statement (page 3 of the same book) 1.Structure sheaf of a ...
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1answer
42 views

Dimension of tensor product vs. fiber product

This is a very fuzzy and ill-formed question, but: we usually think about the tensor product having dimension equal to the product of the dimensions of its factors (at an intuitive level, e.g. vector ...