All Questions
Tagged with divisors projective-varieties
10 questions
3
votes
1
answer
248
views
A question on "Ample subvarieties of algebraic varieties"
Corollary 3.3 in Chapter IV of "Ample subvarieties of algebraic varieties" by R. Hartshorne asserts the following:
Let $X$ be a smooth projective variety and $Y\subset X$ a smooth subvariety ...
- 1,306
2
votes
1
answer
276
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Varieties with disjoint prime divisors
I've been following the works of Totaro, Pereira, and Bogomolov/Pirutka/Silberstein about algebraic varieties over complex numbers with families of disjoint divisors. The last one generalizes results ...
- 66.3k
3
votes
0
answers
121
views
Detecting non-principal Weil divisors on normal varieties using curves
Let $X$ be a normal projective variety over an algebraically closed field $k$. Given any morphism $f:Y\to X$, there is a pullback homomorphism $f^*:\text{Cl}(X)\to\text{Cl}(Y)$, where $\text{Cl}(X)$ ...
- 66.3k
5
votes
1
answer
463
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Square root of a line bundle up to a finite surjective morphism
Given a projective variety $X$ over a field of any characteristic, consider a line bundle $\mathcal{L}$ over $X$.
The existence of a line bundle $\mathcal{L}^\prime$ with an isomorphism ${\mathcal{L}^...
- 39.3k
6
votes
0
answers
490
views
Global sections of canonical line bundle on projective curve with everywhere vanishing derivative
Let $k$ be an algebraically closed field of positive characteristic $p$, $C$ be a curve (projective, non-singular, connected) of genus $g\geq 2$ over $k$ and $\omega \in H^0(C, \Omega_C)$ be a regular ...
- 445
0
votes
0
answers
285
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Pencil of divisors in algebraic geometry
Let $X \subset \mathbb{P}^n$ be projective variety over alg closed field of char $0$ and
$C = V(F), D= V(G) \subset X$ two distinct divisors (e.g. two quadrics,
curves or lines lying in a surface,...) ...
- 5,966
5
votes
1
answer
243
views
existence of birational morphism and divisors
The following result was metioned in a lecture: A nonsingular (or smooth) projective surface (variety of dimension 2) has a
birational morphism to the projective plane, if and only if there
exists an ...
- 887
0
votes
1
answer
281
views
Question about Correspondences from Mumford’s Complex Projective Varieties
I study David Mumford's Algebraic Geometry I - Complex Projective Varieties
and have some problems to understand a step in the proof of Lemma 6.7 (b).
Firstly, the general setting & preparations ...
- 7,320
0
votes
1
answer
131
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On relating $l(A), l(B)$ and $l(A+B)$ for Weil divisors on a smooth projective curve where one of the divisors is effective
Let $X$ be a smooth projective curve over an Algebraically closed field $k$. Let $k(X)$ denote its function field.
If $A, B$ are Weil divisors on $X$ such that $A$ is effective (i.e. $A\ge 0$) , then ...
- 149k
3
votes
0
answers
197
views
Existence of regular hypersurface sections
Let $X$ be a irreducible regular projective variety over $Spec(O_K)$ for some number field $K$. Is it known that there exists at least one hypersurface over $Spec(O_K)$ such that cuts $X$ in a regular ...
- 5,901