Questions tagged [divisors]
For questions related to divisors in the sense of algebraic geometry (Cartier divisors, Weil divisors and so on). For question on divisors in the number theoretic sense please use the tag divisors-multiples.
16 questions
19
votes
2
answers
8k
views
The canonical line bundle of a normal variety
I have heard that the canonical divisor can be defined on a normal variety X since the smooth locus has codimension 2. Then, I have heard as well that for ANY algebraic variety such that the canonical ...
- 2,215
29
votes
7
answers
7k
views
Elementary proof of Riemann-Roch for compact Riemann surfaces
I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps ...
user100272
16
votes
3
answers
4k
views
Contracting divisors to a point
This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up.
If $X$ is a projective variety over an algebraically closed field ...
- 4,450
7
votes
1
answer
861
views
Is $n=6$ the only integer satisfies ${\sigma}_x(n) \equiv 0\bmod{n}$ for every odd integer $x > 0$ and $2 (\bmod n)$ if $x$ is even integer? [closed]
After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ i got this result : $\sigma_x(6)=0 \bmod 6$ for $...
- 414
15
votes
6
answers
2k
views
Seeking Noetherian normal domain with vanishing Picard group but not a UFD
Once again, the question says it all.
My motivation is the article on factorization I am writing. I want to explain (as well as to understand!) why for normal Noetherian domains of dimension greater ...
- 65.4k
10
votes
2
answers
1k
views
Picard group of a cubic hypersurface
Consider the following cubic hypersurface in $\mathbb{P}^5$:
$$
X = \{z_0z_3z_5-z_1^2z_5-z_0z_4^2+2z_1z_2z_4-z_2^2z_3 = 0\}\subset\mathbb{P}^5
$$
The singular locus of $X$ is the Veronese surface $V\...
user168611
7
votes
1
answer
543
views
A constructive proof of the theorem of the cube
Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...
- 1,833
6
votes
0
answers
535
views
When is $ \sigma(n!-1) $ a perfect square?
I am looking for pairs of positive integers $(m,n)$ such that $ \sigma(n!-1) =m^2$, where $\sigma$ is the sum of divisors function. Examples occur with $(m,n)=(12,5),(1,2)$.
Question: Are there ...
user99666
6
votes
1
answer
3k
views
Top self-intersection of exceptional divisors
Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
user49214
3
votes
4
answers
3k
views
Cone over the Veronese surface
Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK_X = \mathcal{O}_X(...
user56259
3
votes
1
answer
437
views
Question about canonical DM stacks
Let $\mathcal{X}$ be a canonical stack (edit: I forgot to say I also want $\mathcal{X}$ smooth), and $\pi : \mathcal{X}\to X$ its coarse moduli space morphism. Let also $D$ be a prime divisor (i.e. ...
- 23.4k
3
votes
0
answers
375
views
Linear system on singular plane curve
Let $C \subset \mathbb{P}^2_k$ an irreducible plane curve of degree $d >1$
over algebraically closed field $k$. That is $C=V(f(x,y,z))$ where $f \in k[x,y,z]$ homogeneous of degree $d$. Let $\{...
- 5,966
3
votes
2
answers
924
views
Rational maps and Kodaira dimension
Let $\phi:X\dashrightarrow Y$ be a generically finite, dominant rational map between smooth projective varieties over $\mathbb{C}$.
Assume that $Y$ is of general type. May we conclude then that $X$ ...
- 8,998
2
votes
1
answer
546
views
divisors and powers of line bundles
Can anyone help me with the following question? Let $X$ be a smooth, projective algebraic variety over a field $k$ of characteristic zero. Let $D$ be an effective divisor on $X$ and $m \geq 2$ an ...
- 23
2
votes
1
answer
488
views
Strict transform under resolution of singularity along a singular $\mathbb{Q}$-Cartier divisor
$\DeclareMathOperator\Bl{Bl}$Let $f: Y=\Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong ...
- 41
2
votes
1
answer
398
views
Picard group of $\mathrm{GL}(n)$-orbits
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group
$$
\GL(n) = \left\lbrace
\left(\begin{array}{cc}
A & C \\
M & B
\end{array}\right) \text{ with } A\...
user168611