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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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 $L=\mathcal{O}_X(\Theta + D)$, ...
TartagliaTriangle's user avatar
1 vote
0 answers
86 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 ...
Dimitri Koshelev's user avatar
5 votes
1 answer
452 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 $\mathcal{O}(K_S)$ ...
user267839's user avatar
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2 votes
0 answers
136 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 $\...
Li Li's user avatar
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3 votes
0 answers
347 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 $\{...
user267839's user avatar
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2 votes
0 answers
147 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 \,\,...
Trusio's user avatar
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3 votes
0 answers
129 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:...
user avatar
1 vote
1 answer
192 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 $p\in X$....
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2 votes
0 answers
420 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 ...
TartagliaTriangle's user avatar
1 vote
0 answers
182 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 ...
TartagliaTriangle's user avatar
1 vote
1 answer
337 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 ...
Jon Aycock's user avatar
2 votes
0 answers
160 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 $X$ ...
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3 votes
0 answers
121 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 $\...
MKR's user avatar
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10 votes
0 answers
212 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 ...
sawdada's user avatar
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2 votes
0 answers
153 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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116 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 ...
pozio's user avatar
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0 votes
0 answers
207 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) ...
user45397's user avatar
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2 votes
1 answer
157 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 = \...
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1 vote
0 answers
120 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: ...
user267839's user avatar
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2 votes
0 answers
131 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}...
Kapil's user avatar
  • 1,536
-1 votes
1 answer
864 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 closed ...
user267839's user avatar
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1 vote
0 answers
111 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: ...
Sheng Meng's user avatar
8 votes
0 answers
308 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 ...
Katharina Hübner's user avatar
3 votes
0 answers
180 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 ...
user127776's user avatar
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3 votes
1 answer
239 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) ...
Francesco Polizzi's user avatar
3 votes
1 answer
297 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 $...
user45397's user avatar
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1 vote
0 answers
104 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 ...
Yeping Zhang's user avatar
3 votes
0 answers
151 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) ...
Andrea's user avatar
  • 263
7 votes
1 answer
1k 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 ...
Joaquín Moraga's user avatar
3 votes
0 answers
204 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 ...
windsheaf's user avatar
  • 435
3 votes
0 answers
360 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 ...
stjc's user avatar
  • 1,072
2 votes
2 answers
481 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, ...
JME's user avatar
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2 votes
0 answers
63 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 ...
Mohsen Karkheiran's user avatar
1 vote
1 answer
739 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 ...
Raffaele C's user avatar
0 votes
0 answers
85 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=...
Stefano's user avatar
  • 625
4 votes
0 answers
205 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 \...
Joaquín Moraga's user avatar
1 vote
1 answer
296 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 $\mathbb{Q}$-...
Stefano's user avatar
  • 625
0 votes
0 answers
147 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 $g+2$. ...
pi_1's user avatar
  • 1,433
7 votes
1 answer
817 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 ...
user43198's user avatar
  • 1,949
8 votes
0 answers
166 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)...
Stefano's user avatar
  • 625
1 vote
0 answers
205 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 ...
user43198's user avatar
  • 1,949
4 votes
0 answers
130 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 ...
user avatar
5 votes
0 answers
497 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....
Li Yutong's user avatar
  • 3,362
3 votes
1 answer
269 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 $...
Li Yutong's user avatar
  • 3,362
3 votes
1 answer
280 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)...
user avatar
2 votes
1 answer
223 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 $Exc(\phi), Exc(\psi)$ ...
user avatar
2 votes
1 answer
203 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 ...
user avatar
14 votes
1 answer
495 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?
user avatar
5 votes
1 answer
232 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 ...
user avatar
4 votes
1 answer
276 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. ...
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