All Questions
Tagged with divisors ag.algebraic-geometry
96 questions with no upvoted or accepted answers
15
votes
0
answers
3k
views
Relative canonical divisors
Suppose that $X$ is a Gorenstein variety and that $\pi : Y \to X$ is a birational map of varieties with normal $Y$.
In this case the relative canonical divisor is defined to be $K_Y - \pi^*K_X$ (if ...
- 20.5k
14
votes
0
answers
3k
views
Ample divisors on projective surfaces
Question: If $X$ is a projective surface and $U$ is an open affine subset of $X$, then is it true that $X \setminus U$ is the support of an (effective) ample divisor on $X$?
Background: I was reading ...
- 5,359
10
votes
0
answers
217
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 ...
- 6,622
8
votes
0
answers
343
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 ...
- 227
8
votes
0
answers
167
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)...
- 625
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
5
votes
0
answers
413
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 ...
- 1,104
5
votes
0
answers
338
views
Distinguishing ample divisors by minimally intersecting curves on a smooth projective toric variety
My question has an easily formulated generalization, which I will state first. Let $\sigma \subseteq \mathbf{R}^n$ be a full-dimensional strongly convex polyhedral cone. For each lattice point $m \in \...
- 197
5
votes
0
answers
540
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....
- 3,472
5
votes
0
answers
232
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 ...
- 625
5
votes
0
answers
248
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 ...
- 1,479
4
votes
0
answers
249
views
Is it always true that the complement of an ample divisor is affine?
Consider a proper and integral scheme $X\rightarrow\operatorname{Spec}(A)$ over a Noetherian ring $A$ and $D\in\operatorname{Div}(X)$ an effective ample Cartier divisor on $X$. Is it true that its ...
- 41
4
votes
0
answers
1k
views
Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?
Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...
- 4,164
4
votes
0
answers
209
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 \...
- 1,595
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 ...
user77919
4
votes
0
answers
172
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 ...
- 625
3
votes
0
answers
138
views
Inverse image Weil divisor on a toric variety as a Cartier divisor
Let $X$ be a normal toric variety over an algebraically closed field and let $D$ be a torus invariant (prime) divisor. Assume $\pi\colon \tilde{X}\rightarrow X$ is a toric resolution of singularities ...
- 337
3
votes
0
answers
405
views
Cartier divisor that is not a difference of two effective Cartier divisors
Note: There are already several related questions, without any definite answer.
I want to find an example of a Noetherian integral scheme $X$ which contains a Cartier divisor that is not linearly ...
- 603
3
votes
0
answers
199
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 ...
user125056
3
votes
0
answers
193
views
branch divisor of this map
We consider the blow up $Bl(\mathbb{P}^2)_p$ of $\mathbb{P}^2$ in $p:=|1:0:0|$ and the following surface:
$Y:=\{(|y_1: y_2:y_3:y_4|, |x_0:x_1:x_2|) \in \mathbb{P}^3\times \mathbb{P}^2: rk(\begin{...
3
votes
0
answers
375
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 $\{...
- 5,986
3
votes
0
answers
135
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:...
user61586
3
votes
0
answers
122
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 $\...
- 93
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
3
votes
0
answers
155
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) ...
- 263
3
votes
0
answers
205
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 ...
- 435
3
votes
0
answers
406
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 ...
- 1,102
3
votes
0
answers
120
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 ...
- 79
3
votes
0
answers
342
views
Hypersurfaces with Gorenstein singular loci
Recall that a hypersurface $D$ in a complex manifold $X$ is called a free divisor if the Lie algebroid $\mathcal{T}_X(-\log D)$ of vector fields tangent to $D$ is locally free. This condition is ...
- 126
2
votes
0
answers
149
views
Non-proper intersection between divisors on $\mathbb{P}^1$-bundle of Hirzebruch surfaces
We are working on algebraic closed field $k$. Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$, $C_0$ and $C_{\infty}$ are its zero and infinity sections ...
- 41
2
votes
0
answers
85
views
Branched covers of real algebraic varieties
Let $X$ be a smooth complex algebraic variety and $L$ be an $n$-torsion line bundle on $X$, i.e., a line bundle $L$ such that $L^n=\mathcal{O}_X(B)$, where $B$ is a divisor $B$ on $X$. Such a bundle ...
- 85
2
votes
0
answers
232
views
Chern classes and rational equivalence
Let $X$ be a complex variety and let $l_1$ and $l_2$ be line bundles on $X$. Let $f_1$ and $f_2$ be sections of $l_1$ and $l_2$ respectively, and let $Z_1$ and $Z_2$ be their zero-sets.
I would like ...
- 89
2
votes
0
answers
244
views
On the definition of the relative canonical divisor
Fix a field $k$. Let $X,Y$ be normal integral $k$-schemes of finite type and $h: Y \to X$ a proper birational $k$-morphism. Moreover, assume that $X$ is Gorenstein, i.e. it admits a canonical divisor $...
- 293
2
votes
0
answers
242
views
Semi-continuity of the Picard number
Let $f:X\rightarrow S$ be a family of smooth projective varieties. For $s\in S$ set $X_s := f^{-1}(s)$, and let $\rho(X_{s})$ be the Picard number of the fiber over $s\in S$. Fix a point $s_0\in S$.
...
- 8,998
2
votes
0
answers
67
views
Irreducible components over a singular divisor
Setup. Let $K$ be an algebraically closed field of characteistic zero, let $X/K$ be a smooth projective surface and let $Z \subset X$ be an integral curve which is nonsingular except for a finite set ...
- 998
2
votes
0
answers
92
views
Linear system of a relative effective divisor on an arithmetic surface contains vertical divisors
I am puzzled by the behavior of some divisors in my attempt to understand the relative Picard functor $\mathrm{Pic}_{X/S}$ of an arithmetic surface $\pi:X\to S$. This is defined by relative divisors $...
- 1,142
2
votes
0
answers
154
views
subspace of the global sections of $\mathcal O$$(D)$
Let $X$ be a smooth projective surface and $D$ an effective divisor whose complete linear system $|D$ is base point free and $D^2=1$. Suppose the dimension of $|D|$ is greater than or equal to 3. Is ...
user169803
2
votes
0
answers
220
views
Divisorial contraction to a non-normal variety
Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which ...
user114666
2
votes
0
answers
142
views
Degree of a divisor along a subscheme
I'm curious about a computation of Prop2.3 in The gonality conjecture on syzygies of algebraic curves of large degree by Ein and Lazarsfeld. Let $C$ be a smooth projective curve carrying a pencil $\...
- 439
2
votes
0
answers
476
views
Uniqueness of theta divisor
Let $A$ be an abelian variety (at least over $\mathbb{C}$). Suppose we have two theta divisors $\Theta_1$ and $\Theta_2$ on $A$, which give two principal polarizations on $A$.
In general, are those ...
2
votes
0
answers
165
views
A question on Okounkov bodies
Let $X$ be an irreducible $n$-dimensional projective variety, and
$$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 $X$ ...
user125056
2
votes
0
answers
163
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?
user61586
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 ...
2
votes
1
answer
491
views
Strict transform under resolution of singularity along a singular $\mathbb{Q}$-Cartier divisor
$\DeclareMathOperator\Bl{Bl}$Let $f: Y=\Bl_0^\omega(\mathbb{C}^3)\to \mathbb{C}^3$ be a weighted blow up of $\mathbb{C}^3$ with weights $w(x,y,z)=(1,1,2)$. Then $Y$ and the exceptional divisor $E\cong ...
- 41
2
votes
0
answers
151
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_{...
- 8,998
2
votes
0
answers
263
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 $\...
- 1,982
2
votes
0
answers
341
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 ...
user82886
2
votes
0
answers
292
views
Psi-classes on moduli spaces of weighted curves
Let $\overline{\mathcal{M}}_{g,A[n]}$ be the stack of weighted genus $g$ curves with weights $A[n]=(a_1,...,a_n)$, and let $\pi:\mathcal{C}\rightarrow \overline{\mathcal{M}}_{g,A[n]}$ be the universal ...
user82886
2
votes
0
answers
112
views
Polarization of the Prym variety
Let $X\rightarrow Y$ a ramifield double cover of curves, $J_X, J_Y$ their jacobians, $P\subset J_X$ the Prym variety, for any line bundle $L$ on $X$ of degree $g_X-1$, denote by $\Theta_L$ the ...
- 1,891
2
votes
0
answers
194
views
A nice way to verify whether the Neron-Severi group of a smooth affine variety is zero
Let $S$ be a smooth affine variety over an algebraically closed field (this could be the field of complex numbers). Is there an 'easy' way to verify whether $NS(S)=0$? Unfortunately, I don't know how ...
- 16.9k