Questions tagged [ag.algebraic-geometry]
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
21,839
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Large integral points on the quadratic twist $ D y^2=x^3+A x +B$
For integers $A,B,D$ and $D$ squarefree let $E : y^2=x^3+A x + B$
and $E_D$ be the quadratic twist of the elliptic curve $E$:
$$ E_D : D y^2=x^3+Ax +B$$
$E_D$ is isomorphic to $ E'_D : y^2=x^3+D^2 A ...
- 24.6k
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Approximation of a continuous curve on commuting matrices
I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that
$[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$....
- 2,970
3
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On the sheaves-functions dictionary
Let $X$ be a variety over a finite field $k$. Let $\pi_{1}(X)$ be the arithmetic etale fundamental group of $X$, and $\rho:\pi_{1}(X)\to k^{\times}$ a continuous character. If $x: \text{Spec}(k)\to X$ ...
- 561
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When $x=\frac{u(f_i,g_j)}{v(f_i,g_j)}$ implies $x=\frac{u(f_i(x,0),g_j(x,0))}{v(f_i(x,0),g_j(x,0))}$ ($x=\frac{xy}{y}$ does not imply $x=\frac{0}{0}$)
Let $f_i=f_i(x,y), g_j=g_j(x,y) \in \mathbb{C}[x,y]$,
$1 \leq i \leq n$, $1 \leq j \leq m$, be such that
$f_i(x,0) \neq 0$ and $g_j(x,0)=0$.
Assume that $\mathbb{C}(f_1,\ldots,f_n,g_1,\ldots,g_m)=\...
- 2,787
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60
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Log resolution and a divisor of pullback of function
Let $(X,x)$ be a three fold singularity
$m_{X,x}$ a ideal sheaf correspoinding to $x$.
$\sigma:Y_1\rightarrow X$ blow up at by $m_{X,x}$
$\phi:Y\rightarrow Y_1$ resolution of $Y_1$
Set $f:=\phi*\sigma$...
- 231
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Infinitely many coprime solutions of $F(x,y)= k(a_1 x + a_2 y)^2 z^2$?
This might be related to an open problem.
Let $F(x,y)$ be homogeneous degree 4 squarefree polynomial
with integer coefficients and
$h(x,y)=a_1 x + a_2 y$ and $\gcd(F,h)=1$ and $k$ be integer.
Consider ...
- 24.6k
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3
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Concrete works by Alexandre Grothendieck, other than Dessin d'Enfants?
For me "Dessin d'Enfants" by Alexandre Grothendieck is the more concrete research work he has done. I would like to know if there are others.
When he was teaching at Montpellier University (...
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Could efficient solutions of $x^2+n y^2=A$ be related to integer factorization?
Let $n$ be positive integer with unknown factorization and $A$ integer with known
factorization.
According to pari/gp developers pari can efficiently find all solutions of:
$$x^2+n y^2=A \qquad (1)$$
...
- 24.6k
5
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Algebraic geometry interpretation for the Calabi metric on the canonical bundle of $CP^n$?
By a construction due to Calabi, there is a special Kahler (=holonomy SU(n+1)) metric in the total space of the canonical line bundle over any constant scalar curvature Kahler manifold. In particular, ...
- 203
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Triple comparison of cohomology in algebraic geometry
Let $X$ be a smooth proper variety over $\mathbb{Q}$ and $p$ a prime number. For an integer $k$, we have:
a finitely-generated abelian group $H^k(X^{\mathrm{an}}(\mathbb{C});\mathbb{Z})$
a finitely-...
- 15.5k
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Integral preimages of topologically noetherian, affine schemes
Let $A\to B$ be a ring homomorphism, $d\in \mathbb{Z}_{>0}$ and let $C=\bigotimes^d_A B$ the $d$-fold tensor product of $B$ over $A$. Then $\mathfrak{S}_d$, the symmetric group of $d$ elements, ...
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$H^2(X,T_X)=0$ implies the Frölicher spectral sequence degenerates at $E_1$?
Let $X$ be a compact complex manifold, if $X$ satisfies $H^2(X,T_X)=0$, then it is well-known that the Kuranishi space of $X$ is smooth by Kodaira and Spencer's deformation theory. On the other hand, ...
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Are there conditions for an elliptic curve to have a quadratic $\mathbb{F}_q$-cover of the line without ramification $\mathbb{F}_q$-points?
Consider an elliptic curve $E: y^2 = f(x) := x^3 + ax + b$ over a finite field $\mathbb{F}_q$ of characteristic $> 3$. Obviously, the projection to $x$ is a quadratic $\mathbb{F}_q$-cover of the ...
- 1,813
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Increase of self-intersection of the canonical divisor on complex smooth projective surfaces
Could anybody tell me where can I find a formula for the increase of the self-intersection of the canonical divisor when, in a smooth complex projective surface, a configuration of curves with ...
2
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1
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140
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Intersection in toric variety
In a toric variety $T$ of dimension $11$ I have a subvariety $W$ of which I would like to compute the dimension.
On $T$ there is a nef but not ample divisor $D$ whose space of sections has dimension $...
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