All Questions
12 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 ...
- 42.9k
19
votes
2
answers
3k
views
Bertini theorems for base-point-free linear systems in positive characteristics
Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least ...
- 1,446
1
vote
1
answer
235
views
Explicit description of the space of global sections of a torus invariant Weil divisor over a real toric variety
Let $D$ be a Weil divisor on a normal toric variety $X$ with fan $\Sigma$ that is invariant under the action by the torus $T$. Then Proposition 4.3.2 of the textbook Toric Varieties by Cox, Little and ...
- 12.9k
7
votes
1
answer
880
views
Bertini's theorem over non-algebraically closed field
Let $K$ be a non-algebraically closed (infinite) field of characteristic $0$ and $X$ a smooth, projective $K$-variety. Does there exist an ample invertible sheaf $\mathcal{L}$ on $X$ such that a ...
- 28.8k
3
votes
0
answers
205
views
Reference request: Vanishing of first cohomology term in Riemann-Roch theorem for singular projective curves over a field
$\newcommand{\F}{\mathcal{F}}$
$\newcommand{\ox}{\mathcal{O}_X}$
Let $f:X \to \operatorname{Spec}(k)$ be a projective scheme of dimension one over a field $k$. The Riemann-Roch equation for such ...
- 435
1
vote
1
answer
324
views
Reference request: $f^*D$ semi-ample, then $D$ semi-ample
I am looking for a suitable reference to put in a bibliography for the following fact:
Let $f: X \rightarrow Y$ be a surjective morphism between normal projective varieties. Let $D$ be a $\mathbb{Q}$-...
- 1,164
7
votes
1
answer
333
views
Pencils on del Pezzo surfaces
Let $X$ be the blow-up of $\mathbb{P}^2$ at three general points $p_1,p_2,p_3$, that is a del Pezzo surface of degree six, and let $\pi_i:X\rightarrow\mathbb{P}^1$ be the morphism induced by the ...
- 39.3k
5
votes
0
answers
248
views
The existence of the Drinfeld shtuka function
I want to understand the existence of the Drinfeld shtuka function but unfortunately I know very little in algebraic geometry.
I am reading Shtukas and Jacobi sums from D. Thakur and I am stucked at ...
- 1,479
2
votes
0
answers
112
views
Polarization of the Prym variety
Let $X\rightarrow Y$ a ramifield double cover of curves, $J_X, J_Y$ their jacobians, $P\subset J_X$ the Prym variety, for any line bundle $L$ on $X$ of degree $g_X-1$, denote by $\Theta_L$ the ...
- 1,891
5
votes
2
answers
3k
views
Embedded resolution of singularities
I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities".
Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" ...
- 42.9k
6
votes
1
answer
868
views
Stable base loci cannot contain isolated points
Let $X$ be a normal projective complex variety.
A theorem of Fujita-Zariski says that if $L$ is a Cartier divisor on $X$ such that
the base locus $Bs(|L|)$ is a finite set then $L$ is semiample.
It ...
- 51
3
votes
4
answers
1k
views
Examples of divisors on an analytical manifold
I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...
- 2,132