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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0answers
85 views

Dimension of a linear system of divisors on singular curve

Consider an singular irreducible plane curve $C \subset \mathbb{P}^2_k$ of degree $d>1$ over algebraically closed field $k$ which is given as vanishing locus $C=V(f(x,y,z))$ of a $f \in k[x,y,z]$ ...
5
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2answers
343 views

Picard group of symplectic group modulo orthogonal group

Let $Sp(2n)$ be the group of complex symplectic $2n\times 2n$ matrices, and $O(2n)$ the group of complex orthogonal $2n\times 2n$ matrices. Consider $Sp(2n)\cap O(2n)\subset Sp(2n)$ and the quotient $...
6
votes
0answers
98 views

Picard group of resolution

Let $X$ be a normal variety and $f:Y\rightarrow X$ a birational morphism, contracting exceptional divisors $E_1,\dots,E_k$ onto the singular locus of $X$, with $Y$ smooth. In this situation is $Pic(Y)...
6
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0answers
112 views

Lefschetz type theorems for linear sections

Let $X\subset\mathbb{P}^n$ be e normal variety, $L\subset\mathbb{P}^n$ a linear subspace, and $Y = X\cap L$ a linear section. Assume that $Y$ is also normal. In particular, we have that $Sing(X)$ has ...
2
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1answer
85 views

Effective semi-group of a singular abelian surface

Let $A$ be a singular abelian surface over $\mathbb{C}$; that is, an abelian surface of maximal Picard rank $\rho(A)=4$. By Shioda-Mitani we know $A \cong E \times E'$ where $E,E'$ are isogenous ...
2
votes
1answer
170 views

Polarization of an abelian variety made by the sum of two divisors

Let $X$ be an abelian variety of dimension $n$, and let $L$ be a polarization, that is, an ample line bundle on $X$, with $\chi(L)=3$. In my specific case, I have that $L=\mathcal{O}_X(\Theta + D)$, ...
1
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0answers
67 views

Picard numbers of isogenous K3 surfaces over a non-closed field

Let $S_1, S_2$ be K3 surfaces defined over a field $k$ and $\phi\!: S_1 \dashrightarrow S_2$ a dominant rational $k$-map (so-called isogeny). It is known that $\rho(S_1) = \rho(S_2)$ for the complex ...
0
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0answers
23 views

Linear system corresponding to a holomorphic embedding from compact Riemann surface to projective space is complete

Le $X$ be a compact Riemann surface and $\phi$ be a holomorphic embedding of $X$ into projective space $\mathbb{C}\mathrm{P}^n$ which is induced by $(f_0,\dots , f_n)$. Then there is a linear system ...
3
votes
1answer
196 views

Residue of the canonical sheaf along subvariety

Let $S$ be a smooth projective surface over an algebraically closed field $k$ and $C \subset S$ a singular curve. Let us denote by $K_S$ the class of canonical divisor of $S$ and $\mathcal{O}(K_S)$ ...
2
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0answers
63 views

Degree of a divisor along a subscheme

I'm curious about a computation of Prop2.3 in The gonality conjecture on syzygies of algebraic curves of large degree by Ein and Lazarsfeld. Let $C$ be a smooth projective curve carrying a pencil $\...
3
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0answers
189 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 $\{...
2
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0answers
88 views

Structure of the big cone and Seshadri constant on Fano manifolds

I would like to know something about the following two questions. Given $X$ Fano manifold and $L$ an ample line bundle on $X$, we define \begin{gather} \sigma(L,x):=\sup\{t>0\, :\, \mu^{*}L-tE \,\,...
4
votes
0answers
99 views

Isomorphisms of weighted complete intersections

Let $X\subset\mathbb{P}(a_0,\dots,a_n)$ and $Y\subset\mathbb{P}(b_0,\dots,b_n)$ be two weighted complete intersections with mild (say terminal) singularities. Assume that there is an isomorphism $f:...
4
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1answer
97 views

Terminal $\mathbb{Q}$-factorial divisorial contractions

Let $X$ be a $3$-fold, and $f:Y\rightarrow X$ a birational $\mathbb{Q}$-factorial divisorial terminal contraction (of relative Picard number one) contracting a divisor $E\subset Y$ to a point $p\in X$....
2
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0answers
134 views

Uniqueness of theta divisor

Let $A$ be an abelian variety (at least over $\mathbb{C}$). Suppose we have two theta divisors $\Theta_1$ and $\Theta_2$ on $A$, which give two principal polarizations on $A$. In general, are those ...
1
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0answers
124 views

Nef divisors on abelian varieties are pullbacks of ample ones

It is well known that for any polarization ( that is, ample line bundle) $L$ on an abelian variety $A$, there is an isogeny $\phi\colon A \to B$ to another abelian variety with a principal ...
1
vote
1answer
114 views

Boundary divisor of projective toroidal compactification

If $F$ is a totally real number field with $[F:\mathbb{Q}] = d>1$, $X$ is the moduli space of Hilbert-Blumenthal Abelian varieties for $F$, and $\overline{X}$ is the projective toroidal ...
3
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0answers
100 views

A question on Okounkov bodies

Let $X$ be an irreducible $n$-dimensional projective variety, and $$Y_n\subset Y_{n-1}\subset\dots\subset Y_1\subset X$$ a flag of irreducible subvarieties such that $Y_i$ has codimension $i$ in $X$ ...
3
votes
0answers
116 views

Extra Algebraic $(1,1)$ cycles on a complex surface

Suppose $x,y,w,z$ are homogeneous coordinates of $\mathbb{CP}^3$, and \begin{eqnarray} X_t := \left(F_t = x f_2 +y g_2 +t F_3 = 0 \right) \end{eqnarray} be a family of degree 3 hypersurfaces in $\...
10
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0answers
198 views

Subvarieties with isomorphic complements

Let $X$ be a smooth irreducible projective variety over $\mathbb C$, $Y_1, Y_2$ are two closed smooth subvarieties. Assume $X-Y_1 \cong X-Y_2$, what do $Y_1$ and $Y_2$ have in common (at least ...
3
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0answers
103 views

Terminal and log canonical singularities

Let $D$ be a divisor with at most terminal singularities in a smooth projective variety $X$. Is the pair $(X,D)$ log canonical?
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0answers
96 views

On the fixed and negative part of a linear system

Let $X$ and $Z$ be smooth complex projective varieties and let $f:X\rightarrow Z$ be a contraction (i.e. $f_\ast\mathcal{O}_X=\mathcal{O}_Z$). Let $F$ be an effective $\mathbb{R}$-divisor on $X$ such ...
0
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0answers
105 views

Local complete intersection and hypersurfaces

Let $Y \subset \mathbb{P}^n$ be a regular, codimension $2$, complete intersection subscheme in $\mathbb{P}^n$ (for example, $Y \cong \mathbb{P}^{n-2}$). Let $X$ be a normal (not necessarily smooth) ...
3
votes
1answer
112 views

Sections of Cartier divisors on toric varieties

Let $X_{\Sigma}$ be a projective toric variety. Consider the total coordinate ring $$S = \mathbb{C}[x_{\rho}\: | \: \rho\in\Sigma(1)]$$ Define $\deg(x_{\rho}) = D_{\rho}$. Now, take a divisor $D = \...
1
vote
0answers
107 views

Question about Local Henselian Rings

I have a question regarding properties/characterizations of local Henselian rings exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces": Here the relevant excerpt: Remark: ...
2
votes
0answers
69 views

Chow group of a pair

In a paper by S. Landsburg the (higher) Chow groups of a pair $(X,Y)$ are defined when $Y$ is a smooth closed subvariety of a smooth variety $X$ as follows. We consider the sub-complex $z^{*}(X;.)_{Y}...
1
vote
1answer
308 views

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 ...
1
vote
0answers
94 views

Iitaka dimension of a $\mathbb{Q}$-Cartier Prime divisor

Let $X$ be a normal projective variety and $D$ a prime divisor such that $mD$ is Cartier for some integer $m>0$. Suppose $H^1(X,\mathcal{O}_X)=0$ and $mD|_D\sim 0$. My questions are the following: ...
7
votes
0answers
145 views

How do I make the components of a Cartier divisor again Cartier divisors?

Let $D$ be an effective Cartier divisor on a normal noetherian scheme $X$. Its irreducible components are codimension $1$ subschemes, i.e. Weil divisors, of $X$ but not necessarily Cartier divisors. I ...
3
votes
0answers
115 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 ...
3
votes
1answer
218 views

Divisibility of a divisor

Let $X$ be a smooth complex projective curve and $f \colon X \to Y$ an étale Galois cover, whose Galois group $G$ is finite and of order $r$. For any $g \in G$, define $$\Delta_g = \{(x, \, g \cdot x) ...
3
votes
1answer
163 views

Effective Cartier divisor is an open property

Let $X$ be a regular affine $\mathbb{C}$-scheme, $A$ a (finitely-generated) $\mathbb{C}$-algebra. Let $Y \subset X \times \mathrm{Spec}(A)$ be a closed subscheme of codimension $1$ such that for each $...
1
vote
0answers
90 views

singular $m$-canonical divisors

[remark for v2] I began by considering curves in v1. I am convinced that the answer is positive. Thanks to Jason Starr and abx. Let $X$ be a complex projective variety. Let $K_X$ be its canonical ...
3
votes
0answers
105 views

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) ...
7
votes
1answer
486 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
0answers
145 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
0answers
166 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
2answers
299 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
0answers
60 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
1answer
371 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
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0answers
76 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
0answers
168 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
1answer
208 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
0answers
134 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
1answer
539 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
0answers
141 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
0answers
95 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
0answers
119 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 ...
5
votes
0answers
267 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
1answer
201 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 $...

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