# 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.

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### Semicontinuity of cohomology of torsion-free sheaves restricted to divisors

Let $X$ be a smooth projective variety, $\mathcal{E}$ a torsion-free coherent sheaf on $X$ and $\mathfrak{d}$ a linear system of divisors in $X$.
I would like to show (at least when $X$ is a surface) ...

**6**

votes

**1**answer

221 views

### Pull-back divisor being Cartier

Let $\pi \colon X \rightarrow Y$ be a projective morphism with connected fibers between normal quasi-projective varieties. Let $N$ be a $\mathbb{Q}$-Cartier divisor on $Y$ so that $\pi^*(N)$ is ...

**3**

votes

**0**answers

120 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 ...

**2**

votes

**0**answers

112 views

### Relative amplitude of the exceptional divisor

Let $f:X'\to X$ be a projective birational morphism between complete algebraic varieties. Assume that the exceptional locus ${\rm Exc}(f)$ is the support of an effective Cartier divisor, can we choose ...

**2**

votes

**2**answers

211 views

### Intuition behind Kawamata's definition of a relative movable Cartier divisor

I am trying to develop a good geometric intuition and to understand the motivation behind Kawamata's definition of a relative movable Cartier divisor in Section 2 of reference [1]:
[1] Y. Kawamata, ...

**2**

votes

**0**answers

58 views

### Blowing up the base of an elliptically fibered (non Weierstrass) threefold

Suppose $X$ is an elliptically fibered smooth threefold, with a nontrivial Mordel-Weil group. Lets call the sections $\sigma_i$ ,$i=1 \dots n$. None of these "sections" are honestly a section, they ...

**1**

vote

**1**answer

180 views

### Direct image of reflexive sheaf via finite, flat map

Suppose $f: X \rightarrow Y$ is a finite, flat (hence locally free) morphism of curves (i.e. schemes of dimension 1, not smooth or even reduced). Suppose $L$ is a reflexive sheaf on $X$, locally free ...

**0**

votes

**0**answers

73 views

### $H$ very ample, $f$ finite, is there uniform $C=C(\mathrm{deg}(f))$ for $C f^* H$ very ample?

Let $X$ and $Y$ be normal projective varieties over $\mathbb{C}$ of dimension $n$. Let $f: X \rightarrow Y$ be a finite morphism. Also, let $H$ be a very ample divisor on $Y$.
Is there a constant $C=...

**4**

votes

**0**answers

140 views

### Birational models and Cartier divisors

Let $X$ be a normal projective variety and $D$ be a Weil divisor on $X$ which is $\mathbb{Q}$-Cartier and Cartier in codimension one.
Can we find a projective birational morphism $\pi\colon Y \...

**1**

vote

**1**answer

137 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 $\mathbb{Q}$-...

**0**

votes

**0**answers

115 views

### Embedding of curves in $\mathbb P^2$

There is a mistake in the following argument but I cannot see where. Can someone help me, please?
Let $C$ be any smooth curve of genus $g\geq 1$ and $D$ a general effective divisor of degree $g+2$. ...

**7**

votes

**1**answer

442 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 ...

**8**

votes

**0**answers

124 views

### On a smooth curve $C$, when is $K_C \sim_\mathbb{Q} (2g-2)P$?

Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property:
There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)...

**1**

vote

**0**answers

86 views

### Twisting a line bundle with the zero section

Let $X$ be a smooth projective curve and $L$ be an invertible sheaf on $X$. Denote by $\mathbb{L}$ the line bundle associated to $L$, $\pi:\mathbb{L} \to X$ the natural morphism and $0_\pi$ the zero ...

**6**

votes

**0**answers

114 views

### Extremal rays in Picard rank two

Let $X$ be a projective variety of Picard rank two. We may assume that $X$ is $\mathbb{Q}$-factorial. Then the Mori cone $NE(X)$ has two extremal rays $R_1,R_2$.
Assume that $R_i$ is generated by ...

**4**

votes

**0**answers

213 views

### When does a Cartier divisor a pull-back of a Cartier divisor?

Suppose $f: Y \to X$ is a projective birational morphism between two varieties with mild singularities. For example, we can assume $X$ is normal and kawamata log terminal, $Y$ is $\mathbb Q$-factorial....

**3**

votes

**1**answer

167 views

### Infinitely small intersections with nef $\mathbb R$-Cartier divisors

Suppose $X/\mathbb C$ is a projective $\mathbb Q$-factorial variety with wild singularities. Let $N$ be a nef $\mathbb R$-Cartier divisor. Then is it possible that there are infinitely many curves $...

**4**

votes

**1**answer

192 views

### Ring of sections and normalization

Let $D$ be a base-point-free divisor on a normal projective variety $X$, and let $Y$ be the image of the morphism $f_{D}:X\rightarrow Y$ induced by $D$. Assume that $f_D$ is birational.
Now, let $X(D)...

**3**

votes

**1**answer

114 views

### Flipping and flipped loci

Let $f:X\dashrightarrow Y$ be the flip of a small contraction $\phi:X\rightarrow Z$, and let $\psi:Y\rightarrow Z$ be the small contraction such that $\psi\circ f = \phi$. Let $Exc(\phi), Exc(\psi)$ ...

**3**

votes

**1**answer

151 views

### Curves contracted by a rational map

Let $D$ be a big but not nef divisor on a normal $\mathbb{Q}$-factorial projective variety. Assume that the section ring
$$R(D) = \bigoplus_{n\in\mathbb{N}}H^0(X,nD)$$
is finitely generated and ...

**15**

votes

**1**answer

292 views

### Birational automorphisms of varieties of Picard number one

Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism.
Must $f$ necessarily contract a divisor?

**6**

votes

**1**answer

151 views

### Blowing-up an ideal generated by squares

Let $f_1,\dots,f_r$ be regular functions on a smooth projective variety $X$, and consider the ideals $I = (f_1^2,\dots,f_r^2)$ and $J = (f_1,\dots,f_r)$. Let $Y = Z(I)$ and $W = Z(J)$ be the ...

**5**

votes

**1**answer

247 views

### Polynomials on spaces of matrices

Let $\mathbb{P}^N$ be the projective space parametrizing $n\times n$ non-zero matrices modulo scalar multiplication, and let $\mathbb{P}^M\subset\mathbb{P}^N$ be the subspaces of symmetric matrices.
...

**8**

votes

**1**answer

214 views

### Push-forward of nef divisors via finite morphisms

Let $f:X\rightarrow Y$ be a finite morphism between smooth projective varieties, and let $D$ be an effective nef but not ample divisor on $X$.
Consider the divisor $f_{*}D$ on $Y$. Is $f_{*}D$ nef ...

**9**

votes

**2**answers

306 views

### Intersection numbers in $\mathbb{P}^1$-bundles

Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence
$$0\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}_p(-1)\...

**5**

votes

**1**answer

273 views

### Anti-canonical divisor of a Fano variety

Let $X$ be a normal projective Fano variety, that is the anti-canonical divisor $-K_X$ is ample.
For any $m>0$ let us consider the complete linear system $|-mK_X|$ and the map
$$f_{|-mK_X|}:X\...

**1**

vote

**0**answers

162 views

### Intersection with very ample divisor and linear equivalence

Let $X$ be a smooth, projective variety and $D, E$ two effective divisors of $X$ which correspond to distinct elements on the cohomology group $H^2(X,\mathbb{Q})$. Denote by $H$ a very ample divisor ...

**4**

votes

**2**answers

199 views

### Is this divisorial contraction a blow-up?

Let $C$ be a curve in a smooth $3$-fold $X$ with an ordinary node $p\in X$. Blow-up $p$ let $E$ be the exceptional divisor, and $\widetilde{C}$ the strict transform of $C$. Furthermore let $L$ be the ...

**6**

votes

**2**answers

290 views

### The kernel of a nef line bundle

Let $V$ be a complex projective variety and $L$ a nef line bundle on $V$ (i.e., $L$ is non-negative on every curve in $V$). Denote, as usual, $\deg_LX = c_1(L)^{\dim{X}}.[X]$ for $X$ a subvariety of $...

**3**

votes

**1**answer

121 views

### Anti-canonical divisorial contractions of weak Fano $3$-folds

Let $X$ be a smooth weak Fano but not Fano $3$-fold ($-K_X$ is nef and big but not ample). Then the anti-canonical morphism $\phi:X\rightarrow W$ (the morphsim induced by the linear system $|-mK_X|$ ...

**7**

votes

**1**answer

179 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 ...

**4**

votes

**1**answer

142 views

### Is the class (resp. Picard) group of a $G$-variety generated by invariant divisors?

Let's work over the complex numbers. Let $S$ be a normal surface, $\mathrm{A}^1(S)$ the class group of divisors on $S$ and $\mathrm{Pic}(S)$ its Picard group. Let $G$ be a reductive group acting on $S$...

**3**

votes

**0**answers

127 views

### Stable base loci and flips

Let $D_1,D_2$ be two effective divisors on o normal and $\mathbb{Q}$-factorial projective variety $X$ of Picard rank two. Assume that $D_1$ is semi-ample and that it induces a small-comtraction $f_{...

**4**

votes

**0**answers

98 views

### A question on the Kodaira dimension of 3-folds

Let $X$ a smooth projective $3$-fold. Assume that $X$ admits a finite rational map $f:X\dashrightarrow Y$ where $Y$ is a smooth Calabi-Yau 3-fold, and a fibration $g:X\rightarrow \mathbb{P}^2$ with a ...

**4**

votes

**0**answers

147 views

### In search for examples concerning pushforward of nef divisors and lc-trivial fibrations

My question is motivated by ideas around the moduli b-divisor of an lc-trivial fibration (see for instance the following paper by Ambro https://arxiv.org/pdf/math/0308143.pdf).
In such a setup, one ...

**30**

votes

**1**answer

1k views

### Are there topological versions of the idea of divisor?

I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead ...

**5**

votes

**1**answer

167 views

### Linear systems on moduli spaces of stable maps

I am studying the general theory of moduli spaces of stable maps, in particuar of the moduli spaces $\overline{M}_{0,n}(\mathbb{P}^r,d)$ of degree $d$ stable maps from a rational curve with $n$ marked ...

**5**

votes

**0**answers

131 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**

vote

**1**answer

416 views

### restriction of a divisor

Suppose I have a Cartier divisor D in a smooth variety X, and suppose I have a subvariety Y in X. Is it always possible to talk about restrict D to Y even when Y might be contained in D? I think it is ...

**3**

votes

**1**answer

166 views

### Kähler metric on compact complex manifolds with simple normal crossing divisor

Let $X$ be a reduced compact complex analytic space of $\dim_{\mathbb{C}}X\ge2$; by [KJ] definition 3.29, remark 3.44 and theorem 3.45, it admits a strong resolution $R(X)$ which is smooth, $E=\pi_X^{-...

**3**

votes

**1**answer

257 views

### Bound on the number of primitive divisors of the $n$th Fibonacci number

It is a result of Carmichael that for any integer $n > 12$, the Fibonacci number $F_n$ has at least one primitive divisor, that is, a prime factor $p$ such that $p$ does not divide any $F_m$ with $...

**3**

votes

**2**answers

279 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 $X$ ...

**2**

votes

**0**answers

171 views

### Global section of line bundle on anti-canonical rational surface

Let $X$ be an anti-canonical rational surface(i.e. $-K_X$ is effective) such that $K_X^2\geq 1$. Let $D$ be a $r$-class divisor ($D^2=r, D^2+D.K_X=-2$, the latter condition can be re-interpreted as $\...

**3**

votes

**1**answer

298 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 ...

**3**

votes

**0**answers

193 views

### Lefschetz type theorems for big and nef divisors

Let $X$ be a smooth projective variety, and $D\subset X$ a smooth nef and big divisor. Assume that the restriction map $Pic(X)\rightarrow Pic(D)$ is an isomorphism over $\mathbb{Q}$.
Under which ...

**2**

votes

**1**answer

160 views

### Divisor class group of quartic surfaces

Let $X\subset\mathbb{P}^3$ be a normal quartic surface with divisor class group $Cl(X)\cong\mathbb{Z}[H]$ generated by the hyperplane section.
What can we say about the singularities of $X$?

**4**

votes

**0**answers

156 views

### How close is $h^0(mD)$ to be a polynomial?

Let $X$ be a normal (or smooth if it helps) projective variety over an algebraically closed field $k$. Fix a Cartier divisor $D$: I am interested in knowing how $h^0(mD)$ behaves as $m$ varies.
At ...

**2**

votes

**0**answers

63 views

### Moving curves and small transformations

Let $f:X\dashrightarrow Y$ be an isomorphism in codimension one between smooth projective varieties. Let $C\subset X$ a curve generating an extremal ray of the cone of moving curves $Mov_1(X)$, and ...

**3**

votes

**0**answers

139 views

### Cone of moving curves

Let $X$ be a projective variety and $C\subset X$ be a moving curve, that is the curves numerically equivalent to $C$ cover a dense open subset of $X$.
How can we detect when $C$ is an extremal ray ...

**8**

votes

**1**answer

255 views

### Degree of equations of secant varieties of Veronese varieties

Let $Sec_r(V)$ be the $r$-secant variety of a Veronse variety $V\subset\mathbb{P}^N$, that is
$$Sec_r(V) = \bigcup_{p_1,...,p_r\in V}\left\langle p_1,...,p_r\right\rangle\subset\mathbb{P}^N$$
where $...