# 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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### dimension of linear system and multiplicity at a point

I recently encountered the following statement which I am unable to prove. Let $X$ be a smooth projective surface and let $L$ be a line bundle on $X$. For $x\in X$ if $h^0(|L|)\geq\frac{m(m+1)}{2}$ ...
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I am trying to calculate some intersection numbers and would appreciate help on the following problem: Consider two divisors $D_1$ and $D_2$. Blowing up their intersection yields $\varphi^{*}(D_i) = \... 1answer 204 views ### Pullback of$\mathbb{R}$-Cartier divisors I am reading the recent book by Kawamata, Algebraic Varieties: Minimal Models and Finite Generation. There is an English translation here . In the bottom of page 16 he says that an$\mathbb{R}$-... 0answers 131 views ### Adjunction formula for non compact surfaces Let$M$be a non compact complex surface and S an embedded compact Riemann surface in$M$. I already know how to show the following equality of fiber bundle: $$\Omega^2_{M}|S =\Omega^1_S \otimes N^*_S... 1answer 224 views ### Volume of a divisor on a smooth projective surface Let X be a smooth projective surface (over complex numbers). Let D be a divisor on X. Then we know that its volume is defined as$$\text{vol}_X(D):= \lim \sup_{m \rightarrow \infty} \frac{h^0(X,... 0answers 92 views ### Some possible implication of existence of a$g^r_d$on a smooth plane curve Let$X$be a smooth plane curve of degree$d$and genus$g$(over complex numbers). For example we can take for the time-being$d=6$and$g=10$. Let's also assume that there exists a divisor on$X$... 0answers 71 views ### Determinantal representation of joins Let$X^n = Z(I_X)\subset\mathbb{P}^N$be an$n$-dimensional irreducible and non degenerate variety. Consider a linear subspace$H = Z(I_H)\subset\mathbb{P}^N$of dimension$N-n-2$disjoint from$X$. ... 1answer 181 views ### Set theoretic equation for Veronese varieties Consider the embedding$f:\mathbb{P}^n\rightarrow\mathbb{P}^N$induced by the complete linear system of degree$d$hypersurfaces of$\mathbb{P}^n$. Its image$V_{n,\,d}$is degree$d$Veronese variety ... 1answer 296 views ### Computations of divisor class monoids Let me first recall some definitions from the very first pages of Bourbaki, Commutative Algebra, Chapter 7, "Divisors". Let$A$be a (commutative) domain,$K$its field of fractions. A ... 1answer 59 views ### What are conditions such that the polynomial$x^2+1$divides$p(y)+q(z)+ax+b=F(x,\, y, \,z)$? [closed] I came across the following problem: What are conditions such that the polynomial$x^2+1$divides$p(y)+q(z)+ax+b=F(x,\,y,\,z)$,? Here$p$and$q$are also polynomials and$a$,$b$are real numbers.... 0answers 104 views ### 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 ... 0answers 132 views ### 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,...) ... 1answer 120 views ### 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$((A,B),Z)\mapsto AZB^{T}$. ... 0answers 272 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 ... 0answers 67 views ### Strict transforms of higher codimension subvarieties The following question could be posed in a more general context but for simplicity I will stick to a particular case. Let$X\subset\mathbb{P}^9$be a smooth variety of degree$d$and dimension$6$. ... 1answer 298 views ### Picard group of$\mathrm{GL}(n)$-orbits$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Mat{Mat}$Consider the general linear group $$\GL(n) = \left\lbrace \left(\begin{array}{cc} A & C \\ M & B \end{array}\right) \text{ with } A\... 2answers 572 views ### Picard group of a cubic hypersurface Consider the following cubic hypersurface in \mathbb{P}^5:$$ 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 $$The singular locus of X is the Veronese surface V\... 0answers 104 views ### subspace of the global sections of \mathcal O$$(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 ...
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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 ...
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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 ...
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### Is the positive part of the Zariski decomposition of a big $\mathbb{R}$-divisor big?

I can't understand why the positive part of the Zariski decomposition of a big class is itself big. More concretely: let $X$ be a smooth projective surface over $\mathbb{C}$. Let $N^1_{\mathbb{R}}(X)$...
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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 ...
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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 ...
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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 $k$ ...
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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 $f \in k[x,y,z]$ ...
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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 $p\in X$....
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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 ...
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 ...
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 ...