All Questions
2,033 questions
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156
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A stalk criterion for unit map to be an isomorphism on étale site
Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...
- 6,048
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votes
1
answer
127
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 434
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votes
1
answer
249
views
On zeros of real polynomials in two variables
Let $P(x,y)$ be a polynomial with real coefficients in two real variables $x,y$ such that the set of zeros of $P(x,y)$ is the real conic curve $Q(x,y)=0$. Will it be true that there exists a ...
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1
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1k
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Explicit bijection between Azumaya algebras and Brauer-Severi schemes
This is kind of the relative version of this question. Even though I made extensive enquiries, I couldn't find good references for this and it seems to me that these questions are pretty well ...
- 1,124
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1
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875
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Semicontinuity of degree of fibers for a proper map
Let $f:X \rightarrow T$ be a proper morphism from a projective scheme $X$ to a smooth projective curve $T$ over $\mathbb{C}$. I know that fiber dimension is upper-semicontinuous, but is the degree of ...
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50
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k specific prime factors guess and related prime guess [duplicate]
there is no more than one group
of continuous composite sequence
of length k composed of only k different specific prime factors.
for example 2 3 5[8 9 10]just only one group. I have prove that k ...
- 29
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303
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For which monic irreducible $f \in \mathbb{C}[x,y][T]$, $\mathbb{C}[x,y][T]/(f)$ is a UFD?
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic irreducible polynomial:
$f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$,
$a_j \in \mathbb{C}[x,y]$, $0 \leq j \leq n-1$.
Denote $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}...
- 2,837
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213
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For which $f \in \mathbb{C}[x,y][T]$, all irreducible elements of $\mathbb{C}[x,y]$ remain irreducible in $\mathbb{C}[x,y,T]/(f)$
Let $f=f(T) \in \mathbb{C}[x,y][T]$ be a monic polynomial:
$f=T^n+a_{n-1}T^{n-1}+\cdots+a_1T+a_0$,
$a_j \in \mathbb{C}[x,y]$.
Denote: $A=\mathbb{C}[x,y]$ and $B=\mathbb{C}[x,y,T]/(f)=\mathbb{C}[x,y][...
- 2,837
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0
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146
views
Limit of a sequence of smooth varieties in Hilbert scheme
Let $\{Z_i\}_{i=1}^\infty$ be a sequence of smooth irreducible $k$-dimensional submanifolds of $\mathbb{C}\mathbb{P}^n$ which converges to a closed subscheme $Z$ in the sense of the Hilbert scheme of $...
- 21.8k
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308
views
Faithfully flat etale morphism from strictly Henselian ring (from Etale Cohomology and the Weil Conjecture by Freitag/Kiehl)
I have question about a statement found in Etale Cohomology and the Weil Conjecture by Freitag, Kiehl at the end of page 15.
It starts with the Remark 1.18 : Let $A$ be a strictly Henselian ring (i.e. ...
- 6,048
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163
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Question about the statement of the going-down theorem of Cohen-Seidenberg in Mumford
In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows.
Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that ...
- 3,625
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129
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Do many homogeneous polynomials help in faster integer root extraction?
Given $n$ homogeneous algebraically independent total degree $2$ polynomials with no $x_1^2,\dots,x_n^2$ variable in $\mathbb Z[x_1,\dots,x_n]$ with promise that it has non-zero integer roots with ...
- 1,826
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votes
3
answers
930
views
Names of certain surfaces
Are there any generally used names for the following algebraic and nonalgebraic surfaces? Any references to literature where the surfaces are studied are also appreciated.
Surface I. Implicit ...
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121
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Reference request: smooth affine curves are planar
Let $X\rightarrow\mathrm{Spec}\:\mathbb{C}$ be an affine smooth morphism of relative dimension$\leq 1$. What is a reference for the fact that there exists a $\mathbb{C}$-locally closed immersion $X\...
user142076
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1
answer
307
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Glue DVR to itself, get a separated non-affine scheme
Here is what seems to be a fun little exercise in algebraic geometry. Take a DVR $R$ and automorphism of $Frac(R)$; glue $Spec(R)$ to itself via this automorphism. Can the glued scheme be separated ...
- 25
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96
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Elementary constraints for the solutions of $z^{n-2}y(y+z)=x^n$?
Related to FLT and this question.
For natural $n > 4 $ define the curve $C_n : z^{n-2}y(y+z)=x^n$.
$C_n$ has the trivial points with $x=0$ for all $n$.
The answer in the linked question shows ...
- 25.4k
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votes
1
answer
269
views
Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals
If a commutative ring with unity has finite Krull dimension, then it satisfies a.c.c. and d.c.c. on prime ideals. The converse is not true in general, as can be seen from here An infinite dimensional ...
- 1,209
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1
answer
2k
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Are Chow groups a birational invariant?
Let us work in the category of smooth, projective varieties (say, over an algebraically closed field $k$). If $X$ and $X'$ are birational, then do they have the same Chow groups? Is there at least a ...
- 179
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123
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A quantity associated with an algebraic variete
Let $P:\mathbb{C}^{n}\to \mathbb{C}$ be an irreducible homogenous polynomial.
Is there a geometric or algebra geometric interpretation for the following quantity:
The maximum number $k$ such that ...
- 366
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0
answers
78
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Direct image of sheaves for blowing ups [duplicate]
Let $X$ be a smooth algebraic variety, $Z\subset X$ a smooth closed subvariety, and
$\pi:\tilde{X}\to X$ the blowing up of $X$ along $Z$. Let $E\subset\tilde{X}$ be the exceptional divisor, and $n$ an ...
- 739
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0
answers
107
views
Cubic monic polynomial over z_p
Let
$$
f_{a}(x)=x^3+(u-2-a)x^2+ax+1,
$$
where $u\in\mathbb{Z}_p^*$ is fixed. Let $S$ be the set consisting of all $a\in\mathbb{Z}_p$ such that $f_{a}(x)$ factor linearly. Then what is the cardinality ...
user126352
0
votes
1
answer
270
views
Proof of rigidity lemma
I have problems to understand a proof in this paper by Pierrick Dartois on Abelian varieties:
Theorem 1.13 (rigidity lemma). Let $ \varphi: X \times_k Y \to Z$ be a morphism of $k$-schemes. Assume ...
- 6,048
0
votes
1
answer
154
views
Dimension of a similarity class
Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...
- 3,741
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votes
0
answers
136
views
Trivializing covers of $\ell$- torsion of elliptic curve
Let $E \to \mathbb{G}_m/k$ an elliptic curve over $ \mathbb{G}_m$ ($k$ field of char $p>0$) and $E[\ell]$ for $(\ell,p)=1$ the $\ell$-torsion group.
Let $f:T \to \mathbb{G}_m$ an finite etale ...
- 6,048
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votes
1
answer
131
views
curve through a point avoiding an hypersurface
Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, $n$ big enough over $\mathbb{C}$. Let $U$ be the complementary open subset.
Let $x\in H$, Is it possible to find an curve $C\subset\mathbb{A}^{...
- 3,472
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votes
1
answer
157
views
For given bundle $E\to X$ find $\xi\to Y$ such that $\pi_*ch(\xi)=ch(E)$
Let $\pi:X\to Y$ a projective morphism and $F\to X$ a vector bundle. The Grothendieck-Riemann-Roch theorem states that
$$\pi_*(ch(F)td(\pi))=ch(\pi_!F)$$
where $td(\pi)$ denotes the relative Todd ...
- 789
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votes
1
answer
860
views
Sierpinski Triangle and the Chaos Game
The chaos game is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave ...
- 87
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1
answer
311
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Intuitive explanation of concentration of the measure for spheres [duplicate]
What is the concentration of the measure(c.o.m.)?
I am struggling with the following sentence;
"The phenomenon of the concentration of the measure for spheres in dimensions larger than 2."
I tried ...
- 11
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1
answer
3k
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Can one expect the existence of a relevant approach for a proof of the Riemann hypothesis using Mochizuki's theory? [closed]
Next month at Oxford university, there will have the first workshop outside Asia on the Inter-Universal Teichmuller theory of Shinichi Mochizuki: http://www.claymath.org/events/iut-theory-shinichi-...
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1
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1k
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Can one embedd the projectivezed tangent space of CP^2 in a projective space?
Given a complex vector bundle $V\rightarrow M$, we can form a
fibre bundle $\mathbb{P} V\rightarrow M$, where the fiber over
each point is the corresponding projective space. In particular
consider ...
- 3,245
-1
votes
1
answer
342
views
Finite or polynomial number integral points clarification on Coppersmith's theorems (possibility of ellipse counter example?)
Coppersmith states if $f(x,y)$ is an irreducible bivariate with total degree $\delta$ then if he can list all roots $(X,Y)$ of the polynomial in $\mathsf{poly}(\log D,\delta)$ time if the roots ...
- 13.9k
-1
votes
1
answer
315
views
Bounds for the number of points on projective hyperelliptic curves over finite fields
Let $C$ be projective hyperelliptic curve over finite field $K$.
What are bounds for the number of points $\#C(K)$?
The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are
not smooth ...
- 25.4k
-2
votes
1
answer
215
views
Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ [closed]
I am reading from the book Topics in Galois theory by Serre.
I have the following question ,
take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by
$$\sigma x\;=\;1/(1-x)$$
where $\sigma$ ...
- 601