# 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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### Is the reduced scheme associated to a Cartier divisor always Cartier?

Let $X$ be a normal integral variety over $\mathbb{C}$ and $D \subset X$ be a Cartier divisor in $X$. Is the associated reduced scheme $D_{\mathrm{red}}$ also necessarily a Cartier divisor in $X$?
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### A constructive proof of the theorem of the cube

Do you know a constructive proof of the theorem of the cube ? More precisely, let $X$, $Y$, $Z$ be projective varieties (e.g., over an algebraically closed field $k$) with points $x$, $y$, $z$ ...
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### Two conditions on divisors on surfaces

Let $X$ be a smooth projective surface and $D$ be an effective Cartier divisor (not necessarily ample) on $X$. Is there a connection between these two conditions? $(i)$ for a large enough $n$, the ...
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### Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve

Consider an elliptic curve $E \subset \mathbb{P}^2$ with the zero point $\mathcal{O}$. There are classical articles about complete systems of addition laws on $E$ (see Lange and Ruppert - Complete ...
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$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$Let $X$ be a compact complex manifold, assume projective if you'd like. Define the Néron–Severi group to be the quotient $$\NS(X) = \Pic(X) / \... 6 votes 0 answers 222 views ### Global sections of canonical line bundle on projective curve with everywhere vanishing derivative Let k be an algebraically closed field of positive characteristic p, C be a curve (projective, non-singular, connected) of genus g\geq 2 over k and \omega \in H^0(C, \Omega_C) be a regular ... 2 votes 1 answer 144 views ### Restriction of small transformations Let \phi:X\dashrightarrow Y be an elementary small transformation (isomorphism in codimension 1) between normal and \mathbb{Q}-factorial projective varieties. Then there are small contractions ... 2 votes 0 answers 80 views ### principal divisor on complex surfaces Let X be a non compact complex surface non projective and non algebraic, and let S be compact Riemann surface embedded in X ( i mean that S is a compact complex sub variety of X of ... 0 votes 0 answers 102 views ### Intersection product when one factor is contained in the exceptional divisor I am trying to calculate some intersection numbers and would appreciate help on the following problem: Consider two divisors D_1 and D_2. Blowing up their intersection yields \varphi^{*}(D_i) = \... 3 votes 1 answer 266 views ### Pullback of \mathbb{R}-Cartier divisors I am reading the recent book by Kawamata, Algebraic Varieties: Minimal Models and Finite Generation. There is an English translation here . In the bottom of page 16 he says that an \mathbb{R}-... 0 votes 0 answers 148 views ### Adjunction formula for non compact surfaces Let M be a non compact complex surface and S an embedded compact Riemann surface in M. I already know how to show the following equality of fiber bundle:$$\Omega^2_{M}|S =\Omega^1_S \otimes N^*_S$... 5 votes 1 answer 320 views ### Volume of a divisor on a smooth projective surface Let$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,... 9 votes 1 answer 247 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 ... 5 votes 1 answer 330 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 ... 0 votes 1 answer 68 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.... 0 votes 0 answers 129 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 ... 0 votes 0 answers 165 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,...) ... 3 votes 1 answer 138 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}. ... 5 votes 0 answers 309 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 ... 2 votes 1 answer 334 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\... 9 votes 2 answers 764 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\... 113 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 ... 119 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 ... 189 views

### 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 ... ### 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 $N^1_{\mathbb{R}}(X)$...
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\... 