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.
334
questions
-1
votes
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864
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Restriction of a Cartier divisor
Let $X$ be a surface (so $2$-dimensional proper $k$-scheme)
$D \subset X$ an effective Cartier divisor of $X$ which corresponding to an invertible sheaf $\mathcal{L}=O_X(D)$ and
$C \subset X$ a closed ...
2
votes
0
answers
52
views
Conditions for long exact sequence for line bundles on curve to degenerate?
Let $\varphi:X\to Y$ be a morphism of schemes of relative dimension 1, and $\mathcal{L}' \xrightarrow{g} \mathcal{L}$ an injection of line bundles on $X$.
The sequence
$$0\to \mathcal{L}' \xrightarrow{...
4
votes
1
answer
218
views
Volume of conic bundles
Consider a smooth conic bundle $X\rightarrow \mathbb{P}^1$ with discriminant of degree $d$ (the locus of $\mathbb{P}^1$ over which the fibers are reducible conics). There is a formula for $(-K_X)^2$ ...
4
votes
1
answer
347
views
Zsigmondy's Theorem Generalization
Zsigmondy's Theorem states that if $a>b>0$ are coprime integers then for any integer $n\geq 1$ there is a prime $p$ that divides $a^n-b^n$ and does not divide $a^k-b^k$ for any positive integer $...
2
votes
0
answers
120
views
Average length of consecutive integers which have an increasing number of divisors
Consider the nine consecutive natural numbers starting from $1584614377$.
...
3
votes
0
answers
102
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)$ ...
6
votes
1
answer
581
views
Hartshorne's proof of Halphen's theorem
Apologies if this is not quite at the level of MathOverflow, but it has already been asked at MSE and gone unresolved for several years despite a bounty.
Hartshorne states the theorem as follows:
...
1
vote
0
answers
48
views
Torsion order on Prym variety
Consider two hyperlliptic curves $C_1,C_2$ over $\mathbb{Q}$, and a morphism $\phi:C_1 \rightarrow C_2$. Lifting this morphism on the Jacobians of $C_1,C_2$ and taking its kernel defines a Prym ...
11
votes
1
answer
851
views
Is the divisibility graph of the proper divisors of n more often planar than not?
Define the divisibility graph of a set of positive integers as the graph whose vertices are the integers, two of which are joined by an edge if one divides the other.
For all N, is it true that ...
5
votes
1
answer
405
views
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}^...
2
votes
1
answer
191
views
Existence of terminal $3$-fold flips
Does there exist a terminal $3$-fold $X$ with a curve $C\subset X$ such that $K_X\cdot C < 0$ admitting a Mori flip $X\dashrightarrow Y$, flipping $C$ to a curve $C'\subset Y$, where the singular ...
6
votes
2
answers
380
views
Nef divisors on surfaces
Let $X$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $D$ a divisor on $X$ such that $D$ is nef and $D^2 = 0$ with the following properties:
$...
4
votes
1
answer
545
views
Cohomology of divisors on Hirzebruch surfaces
Consider the Hirzebruch surface $\mathbb{F}_n = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus \mathcal{O}_{\mathbb{P}^1}(n))\rightarrow\mathbb{P}^1$. The Picard group of $\mathbb{F}_n$ is generated by ...
3
votes
2
answers
345
views
Abelian varieties corresponding to Hodge substructures
In an exercise of Voisin book, says:
Let $j:C\rightarrow S$ the inclusion of a smooth curve on a smooth connected projective surface. Set
$H=ker(j_*:H^1(C,\mathbb{Z})\rightarrow H^3(S,\mathbb{Z}))$.
...
1
vote
0
answers
196
views
Mori cone of Picard rank two varieties
Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that
$$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$
is an isomorphism, where $i:S\...
3
votes
2
answers
865
views
Rationality of conic bundles
Let $\pi:X\rightarrow\mathbb{P}^2$ be a $3$-fold conic bundle, and let $\Delta\subset\mathbb{P}^2$ be its discriminant. Assume that both $X$ and $\Delta$ are smooth and that $deg(\Delta)\geq 6$.
Can ...
2
votes
1
answer
408
views
Divisors on projective bundles
Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$.
If $n = 0$ then $X = \mathbb{P}^1$, and for $n ...
5
votes
2
answers
498
views
Divisors whose restriction is big
Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$.
...
0
votes
1
answer
173
views
Curves in conic bundles
Consider a smooth minimal $3$-fold conic bundle $f:X\rightarrow\mathbb{P}^2$. Then $X$ has Picard rank two and consequently also the vector space of $1$-cycles is $2$-dimensional. Then the cone of ...
6
votes
2
answers
481
views
Global sections of multiples of a divisor
Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$.
Is there any example where $h^0(X,mD) = 0$ ...
1
vote
0
answers
152
views
Rational classes of $(-2)$-curves in a minimal surface of general type
Let $X$ be a minimal surface of general type over $\mathbb{C}$. One can show that if for any set of $(-2)$-curves $C_1,\cdots,C_l$ on $X$, there exists $k$, $1\le k\le l$ such that $$\sum_{i=1}^k\...
2
votes
1
answer
418
views
A question on effective divisors
Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...
2
votes
1
answer
372
views
Nef and pseudo-effective divisors over non algebraically closed fields
Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective.
...
0
votes
1
answer
253
views
Pseudoeffective divisors on surfaces
Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $...
3
votes
1
answer
465
views
Examples of complex manifolds with trivial Néron–Severi group?
$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$Let $X$ be a compact complex manifold, assume projective if you'd like. Define the Néron–Severi group to be the quotient $$\NS(X) = \Pic(X) / \...
6
votes
0
answers
417
views
Global sections of canonical line bundle on projective curve with everywhere vanishing derivative
Let $k$ be an algebraically closed field of positive characteristic $p$, $C$ be a curve (projective, non-singular, connected) of genus $g\geq 2$ over $k$ and $\omega \in H^0(C, \Omega_C)$ be a regular ...
2
votes
0
answers
128
views
principal divisor on complex surfaces
Let $X$ be a non compact complex surface non projective and non algebraic, and let $S$ be compact Riemann surface embedded in $X$ ( i mean that $S$ is a compact complex sub variety of $X$ of ...
5
votes
1
answer
500
views
Volume of a divisor on a smooth projective surface
Let $X$ be a smooth projective surface (over complex numbers). Let $D$ be a divisor on $X$. Then we know that its volume is defined as $$\text{vol}_X(D):= \lim \sup_{m \rightarrow \infty} \frac{h^0(X,...
0
votes
0
answers
170
views
Intersection product when one factor is contained in the exceptional divisor
I am trying to calculate some intersection numbers and would appreciate help on the following problem:
Consider two divisors $D_1$ and $D_2$. Blowing up their intersection yields $\varphi^{*}(D_i) = \...
3
votes
1
answer
389
views
Pullback of $\mathbb{R}$-Cartier divisors
I am reading the recent book by Kawamata, Algebraic Varieties: Minimal Models and Finite Generation. There is an English translation here .
In the bottom of page 16 he says that an $\mathbb{R}$-...
2
votes
1
answer
343
views
Is the positive part of the Zariski decomposition of a big $\mathbb{R}$-divisor big?
I can't understand why the positive part of the Zariski decomposition of a big class is itself big. More concretely:
let $X$ be a smooth projective surface over $\mathbb{C}$. Let $N^1_{\mathbb{R}}(X)$...
0
votes
0
answers
165
views
Adjunction formula for non compact surfaces
Let $M$ be a non compact complex surface and S an embedded compact Riemann surface in $M$.
I already know how to show the following equality of fiber bundle:
$$\Omega^2_{M}|S =\Omega^1_S \otimes N^*_S$...
3
votes
2
answers
751
views
what are the singularities of a normal crossings divisor?
This is probably a very stupid question. I'm sorry.
Let $D$ be a simple normal crossings divisor on some smooth projective variety $D$. By this I mean that the irreducible components $D_i$ are smooth ...
2
votes
1
answer
496
views
Divisors on the symmetric product of an elliptic curve
Assume that $C$ is an elliptic curve and $C_p$ is the $p$-fold symmetric product. Let $\beta:C_p\to C$ be defined by the addition on the elliptic curve. Let $u\in C$ be the zero in the additive group ...
9
votes
1
answer
383
views
Set theoretic equation for Veronese varieties
Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...
5
votes
1
answer
355
views
Computations of divisor class monoids
Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors".
Let $A$ be a (commutative) domain, $K$ its field of fractions. A ...
0
votes
1
answer
87
views
What are conditions such that the polynomial $x^2+1$ divides $p(y)+q(z)+ax+b=F(x,\, y, \,z)$? [closed]
I came across the following problem:
What are conditions such that the polynomial $x^2+1$ divides $p(y)+q(z)+ax+b=F(x,\,y,\,z)$,?
Here $p$ and $q$ are also polynomials and $a$, $b$ are real numbers....
0
votes
0
answers
212
views
Transversally intersecting divisors $C$ and $D$ in a Hartshorne's AG lemma
Question about proof of lemma V.1.3 in Robin Hartshorne's
Algebraic Geometry on page 358.
Let $X$ be surface. That's for us a nonsingular projective
surface over an algebraically closed field $k$ and ...
0
votes
0
answers
262
views
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,...) ...
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
$$
The singular locus of $X$ is the Veronese surface $V\...
3
votes
1
answer
155
views
Picard group of $(SL(n)\times SL(m))$-orbits
Let $\mathbb{P}^N$ be the projective space of $n\times m$ matrices with complex entries modulo scalar. Consider the $(SL(n)\times SL(m))$-action on $\mathbb{P}^N$ given by $((A,B),Z)\mapsto AZB^{T}$. ...
5
votes
0
answers
370
views
Most divisors on a curve aren't special?
I have a generic smooth curve $C$ of genus $g$ and fixed multiplicities $a_1, \dots, a_n \geq 0$ with $\sum a_i = g+1$.
Q1 : For generic marked points $p_1, \dots, p_n \in C$, must $\sum a_i p_i$ be a ...
2
votes
1
answer
390
views
Picard group of $\mathrm{GL}(n)$-orbits
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group
$$
\GL(n) = \left\lbrace
\left(\begin{array}{cc}
A & C \\
M & B
\end{array}\right) \text{ with } A\...
2
votes
0
answers
140
views
subspace of the global sections of $\mathcal O$$(D)$
Let $X$ be a smooth projective surface and $D$ an effective divisor whose complete linear system $|D$ is base point free and $D^2=1$. Suppose the dimension of $|D|$ is greater than or equal to 3. Is ...
2
votes
0
answers
184
views
Divisorial contraction to a non-normal variety
Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which ...
5
votes
1
answer
223
views
existence of birational morphism and divisors
The following result was metioned in a lecture: A nonsingular (or smooth) projective surface (variety of dimension 2) has a
birational morphism to the projective plane, if and only if there
exists an ...
3
votes
1
answer
187
views
Weak Fano varieties and small transformations
A projective normal and $\mathbb{Q}$-factorial variety $X$ is said to be log Fano if there exists and effective divisor $D$ on $X$ such $-K_X-D$ is ample and the pair $(X,D)$ is klt.
Now, let $f:X\...
1
vote
1
answer
611
views
Push-forward of divisors and intersections
Let $f:X\rightarrow Y$ be a surjective finite morphism of varieties, with $X$ normal and $Y$ smooth. Let $D\subset X$ be a divisor and $C\subset Y$ a curve. Does the equality
$$C\cdot f_{*}D = f^{*}C\...
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(...
3
votes
0
answers
165
views
Divisorial contractions and singularities
I have a smooth $6$-fold $X\subset\mathbb{P}^n$ and a divisor $D\subset X$ cut out by a quadratic polynomial. I know that $D$ in singular along a smooth $3$-fold $Y\subset X$, and that if $Z$ is the ...