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.
345 questions
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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 ...
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285
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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,...) ...
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2
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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\...
- 39.3k
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answer
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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 $((A,B),Z)\mapsto AZB^{T}$. ...
- 14.2k
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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
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Picard group of $\mathrm{GL}(n)$-orbits
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group
$$
\GL(n) = \left\lbrace
\left(\begin{array}{cc}
A & C \\
M & B
\end{array}\right) \text{ with } A\...
- 14.2k
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155
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subspace of the global sections of $\mathcal O$$(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 ...
CommunityBot
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220
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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 ...
- 63.9k
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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 ...
- 887
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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\...
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729
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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
$$C\cdot f_{*}D = f^{*}C\...
- 1,104
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4
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Cone over the Veronese surface
Let $V\subset\mathbb{P}^5$ be the Veronese surface and let $X\subset\mathbb{P}^6$ be the cone over it. Since $X$ is $\mathbb{Q}$-factorial there are two integers $a,b$ such that $aK_X = \mathcal{O}_X(...
- 42.9k
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Divisorial contractions and singularities
I have a smooth $6$-fold $X\subset\mathbb{P}^n$ and a divisor $D\subset X$ cut out by a quadratic polynomial. I know that $D$ in singular along a smooth $3$-fold $Y\subset X$, and that if $Z$ is the ...
CommunityBot
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Generalisations of Riemann-Roch for surfaces
Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have
$$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$
This is the famous ...
- 2,215
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First Chern class and field extensions
Let $X$ be a smooth, complex projective algebraic variety defined over a number field $K$.
Let $D$ be a divisor of $X$ defined over $K$ with the following property:
For any curve $C$ defined over $K$,...
CommunityBot
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vote
1
answer
210
views
Gcd of linear function
$\DeclareMathOperator\gcd{gcd}$Take $q\in \mathbb N$ and $X>0$ ($q$ not necessarily smaller than $X$). A sum such as
$$\sum_{d\leq X}(q,d)$$
is easily seen to be $\ll q^\epsilon (X+q)$ so that the ...
- 105k
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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{...
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171
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Steps of the MMP "in family"
Let $\pi\colon X\to Y$ be a morphism between irreducible varieties with terminal singularities (let us say smooth if you want). I suppose that I have an open subset $U$ of $Y$ over which the fibres ...
- 7,722
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Question about Correspondences from Mumford’s Complex Projective Varieties
I study David Mumford's Algebraic Geometry I - Complex Projective Varieties
and have some problems to understand a step in the proof of Lemma 6.7 (b).
Firstly, the general setting & preparations ...
- 7,320
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464
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Pushforward of a very ample line bundle on a curve to $\mathbb{P}^1$
Let $p:C\to\mathbb{P}^1$ be a degree $k$ morphism from a smooth projective curve $C$ to the projective line and $L$ a very ample line bundle on $C$. We know that $p_*\mathcal{O}_C(L)$ is a rank $k$ ...
- 35.2k
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338
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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 \...
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Line bundles vs. Cartier divisors on a non-integral scheme
It is well-known that if $X$ is an integral scheme, then there is an isomorphism $CaCl(X)\to Pic(X)$ taking $[D]$ to $[\mathcal{O}_X(D)]$. Does anyone know any simple examples where the above map ...
- 4,713
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1
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131
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On relating $l(A), l(B)$ and $l(A+B)$ for Weil divisors on a smooth projective curve where one of the divisors is effective
Let $X$ be a smooth projective curve over an Algebraically closed field $k$. Let $k(X)$ denote its function field.
If $A, B$ are Weil divisors on $X$ such that $A$ is effective (i.e. $A\ge 0$) , then ...
- 149k
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157
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The morphisms induced by two Cartier divisors
Let X be a projective variety. We consider two Cartier divisors $D,E$ such that $E\geq D$ and the relative morphisms
$\phi_D: X - - -> \mathbb{P}(H^0(X, O_X(D))^*)$ and $\phi_E: X- - -> \mathbb{...
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183
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Problem regarding existence of a divisor representing line bundle
We consider a normal irreducible variety $X$ and a line bundle $L$. The question is when $L$ is induced by a Cartier divisor $D$. We know that if $s$ is a rational section of $O_X(D)$, where $D$ is a ...
5
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2
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492
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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 $...
- 14.2k
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0
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212
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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]$ ...
- 63.9k
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375
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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
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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 ...
CommunityBot
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276
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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)$, ...
- 2,427
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135
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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:...
- 1,164
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91
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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 ...
- 1,833
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1
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512
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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)$ ...
- 42.9k
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142
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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
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154
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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 \,\,...
- 71
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1
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215
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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$....
- 8,998
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476
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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 ...
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0
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194
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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
1
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399
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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 ...
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165
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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$ ...
CommunityBot
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122
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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
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217
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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
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163
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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?
CommunityBot
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116
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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 ...
- 625
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228
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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) ...
- 2,323
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1
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160
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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 = \...
- 8,998
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120
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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: ...
- 5,986
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141
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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}...
- 3,486
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0
answers
114
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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:
...
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343
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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 ...
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