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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7
votes
2answers
252 views

Fibrations of projective varieties

Let $f:X\rightarrow Y$ be a flat morphism of normal projective varieties with fibers of positive dimension (in particular all the fibers are connected and of the same dimension). Let $g:X\rightarrow ...
0
votes
1answer
74 views

Degree of the negative part of a divisor

Let $K$ be an algebraically closed field (or $\overline{\mathbb{C}(z)}$ for a more precise condition). And let $P \in K[x,y]$ be an irreducible polynomial of degree $m$ with respect to $x$ and degree ...
1
vote
1answer
164 views

On Q-Cartier Divisors

I have my question on Q-Cartier Weil divisor. People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of ...
2
votes
0answers
95 views

Blow-up of the diagonal

Let $\Delta$ be the small diagonal in $\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1$, and let $X$ be the blow-up of $\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1$ along $\Delta$ with ...
8
votes
1answer
299 views

Blowing-up a point in the singular locus

Let $X\subset\mathbb{P}^n$ be a variety singular along a smooth subvariety $Z\subset X$ of positive dimension. Let us assume that $X$ has ordinary singularities along $Z$. Now, let $\pi:Y\rightarrow ...
0
votes
0answers
66 views

Terminology regarding divisor on a curve

Suppose that $D = \sum n_i P_i$ is a divisor on a curve $C$, say, over a field. Is there a standard algebraic geometry terminology referring to the set $\{ P_i : n_i \neq 0 \} \subset |C|$? Support of ...
3
votes
2answers
207 views

A question on the effective cone

Let $X$ be a projective variety and $G$ a finite group acting on $X$. We consider the quotient $\pi:X\rightarrow Y :=X/G$. I'm interested in the relation between $Eff(X)$ and $Eff(Y)$. In ...
2
votes
1answer
317 views

A question about an intersection number

Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Now, Let $E_p,E_q, E_{p,q}$ be respectively the ...
2
votes
1answer
162 views

Reference request: log Fano varieties

I need a reference for a proof of the following fact: let $X$ be a toric variety then $X$ is log Fano. Thanks a lot.
1
vote
1answer
126 views

Does a line bundle on a normal Noetherian algebraic space come from a Weil divisor?

Let $X$ be a normal Noetherian algebraic space and $\mathscr{L}$ a line bundle on $X$. If $X$ is a scheme, then there is locally principal Weil divisor on $X$ that gives rise to $\mathscr{L}$. Is the ...
0
votes
1answer
111 views

Intersection Matrix of a resolution

Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that $$K_X = f^{*}K_S+\sum_ia_iE_i$$ with $a_i>0$. By Grauert-Mumford theorem the ...
0
votes
1answer
112 views

Schematic image of a relative Cartier divisor of a fiberwise dense open

Let $S$ be a scheme and $A$ an abelian $S$-scheme, i.e., $A \rightarrow S$ is a proper smooth $S$-group scheme whose fibers are $g$-dimensional abelian varieties. Suppose that one has a fiberwise ...
1
vote
1answer
186 views

Moving a divisor on a (reducible, non-reduced) curve

I am trying to understand the first sentence of the proof of 9.1/5 in "Neron models." There we have a proper curve $X$ over a field $K$ and a line bundle $\mathscr{L}$ on $X$. Our ultimate goal is to ...
6
votes
4answers
674 views

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 = ...
2
votes
3answers
311 views

Movable Divisors

Let $X$ be a projective variety. Does anyone know an example of a movable reducible divisor $D\in Mov(X)$ such that any element in the linear system $|D|$ of $D$ is reducible?
8
votes
2answers
359 views

Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$

Let us consider the points $$p_1=[1:0:...:0],p_2 = [0:1:...:0],...,p_{n-2} =[0:...:0:1],\\ p_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$ and the blow-up $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$. ...
6
votes
1answer
527 views

Bertini's Theorem

Let $p_1,...,p_n\in\mathbb{P}^{N}$ be general points. Consider the linear system $|L|$ of hypersurfaces of degree $d$ in $\mathbb{P}^{N}$ with prescribed multiplicities $m_1,...,m_n$ at $p_1,...,p_n$. ...
3
votes
2answers
204 views

Standard plane Cremona transformation

Let us consider nine general points $p_1,...,p_9\in\mathbb{P}^2$ and the line $L = \left\langle p_1,p_2\right\rangle$. Take the standard Cremona $f_1$ centred in $p_3,p_4,p_5$, then $C_1 = f_1(L)$ is ...
1
vote
0answers
110 views

A question on the secondary fan

I am studying the secondary fan decomposition of the effective cone of a projective variety $X$. Let as assume that $X$ is a Mori Dream Space. As far as I understand passing from a cone of maximal ...
4
votes
1answer
135 views

Canonical bundle of moduli space of rational curves and automorphisms

Let $\overline{M}_{0,n}$ be the usual Deligne-Mumford compactification of $M_{0,n}$ the moduli space of smooth $n$-pointed rational curves. The canonical divisor $K_{\overline{M}_{0,n}}$ can be ...
3
votes
1answer
128 views

A question on klt pairs

Let $D$ be a $\mathbb{Q}$-divisor in a smooth variety $X$. In Lazarsfeld book "Positivity in Algebraic Geometry 2" I found Proposition 9.5.13 saying that if for any $x\in D$ we have $mult_xD < 1$ ...
0
votes
1answer
111 views

Global sections in negative degree line bundles over singular curves

Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth ...
2
votes
1answer
174 views

Singularities of secant varieties of rational normal curves

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$, and let $Sec_k(C)\subset\mathbb{P}^n$ be its $k$-th secant variety. By Theorem 1.1 in this paper: ...
1
vote
2answers
204 views

Small birational maps and singularities of the pair

Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair ...
1
vote
1answer
113 views

Boundedness of the number of curves negative on a varying big divisor

For a divisor $D$ on a smooth complex projective surface $X$, the stable fixed part is the maximal effective divisor $E$ which, for every $n \in \mathbb{N}$, is contained in every memeber of the ...
1
vote
1answer
201 views

Cremona transformations

Let $f:\mathbb{P}^n_1\dashrightarrow\mathbb{P}^n_2$ be the standard Cremona transformation based on $p_1,...,p_{n+1}\in\mathbb{P}^n_1$ and $q_1,...,q_{n+1}\in\mathbb{P}^n_2$. That is, $f$ is the ...
1
vote
1answer
127 views

Big divisors and small transformations

Let $X$ be a smooth projective variety such that $-K_X$ is ample. Let $f:X\dashrightarrow Y$ be a small $\mathbb{Q}$-factorial transformation. I would like to know if is true or not that: $-K_Y$ is ...
3
votes
0answers
112 views

Non-vanishing of $H^0$ on rational surface

Let $X$ be a smooth rational surface that admits a proper birational morphism to $\mathbb{P}^2$ and $D$ a simple normal crossing divisor on $X$ such that $K_X+D$ is big, is it true that ...
0
votes
1answer
121 views

Dimension of image of a hyperplane section

If we have a surjective morphism $f:X\to Y$, where $X$ is $n$ dimensional projective variety and $Y$ is $m$ dimensional projective variety. If $m<n$, Can we choose a general hyperplane section $H$ ...
9
votes
2answers
3k views

does a line bundle always have a degree

For curves there is a very simple notion of degree of a line bundle or equivalently of a Weil or Cartier divisor. Even in any projective space $\mathbb P(V)$ divisors are cut out by hypersurfaces ...
1
vote
1answer
148 views

Ample divisors on $\mathbb{P}^n$ blown-up at $k$ general points

Let $X$ be the blow-up of $\mathbb{P}^n$ at $k$ general points. We can assume $k\leq n+4$. Let $$D = aH-b_1E_1-...-b_kE_k$$ be a divisor on $X$. Are there conditions on $a,b_1,...,b_k$ ensuring that ...
3
votes
1answer
190 views

A question about kawamata's proof of vanishing for big and nef $\mathbb{Q}$ divisors

Theorem 2 [1, p.46] Let $X$ be a non-singular projective algebraic variety of dimension $n$, and $D$ a numerically effective $\mathbb{Q}$-divisor such that $(D^n)>0$. We assume that the support of ...
0
votes
1answer
158 views

Singular irreducible quadrics

Let $Q\subset\mathbb{P}^n$ be the quadric hypersurface defined by $$x_0^2+x_1^2+...+x_k^2 =0.$$ If $2\leq k\leq n-1$ then $Q$ is irreducible and $Sing(Q)$ is a linear space of dimension $n-k-1$. If ...
0
votes
1answer
479 views

Simple normal crossing divisors

I found the following definition. A Weil divisor $D = \sum_{i}D_i \subset X$ on a smooth variety $X$ is simple normal crossing if for every point $p \in X$ a local equation of $D$ is ...
4
votes
2answers
297 views

Varieties with big anti-canonical divisor

I recently heard about the following problem: Let $X$ be a projective variety with klt singularities and such that $-K_X$ is big. Is $X$ a Mori Dream Space ? Now, $-K_X$ big if and only if $-K_X ...
0
votes
3answers
318 views

Higher cohomology of sheaves on a projective space

Let $S\subset\mathbb{P}^n$ be a finite set of $s$ reduced points. Let $\mathcal{I}$ be the ideal sheaf of $S$ in $\mathbb{P}^n$. We consider the sheaf ...
1
vote
2answers
483 views

Big and Nef divisors

In Example 2.2.19 of Lazarsfeld, Positivity in Algebraic Geometry I, I found the following statement: Let $D$ be a divisor on an irreducible projective variety $X$. Then $D$ is nef and big if and ...
3
votes
2answers
888 views

(Anti)Canonical divisor of a blow up

This question may be utterly trivial, or not, but as someone with hardly any knowledge of algebraic geometry I thought there could be a chance I get lucky. Let X be a rational surface obtained by n ...
5
votes
1answer
592 views

Top self-intersection of exceptional divisors

Let $Y\subset\mathbb{P}^n$ be a smooth variety of codimension two. Consider the blow-up $X = Bl_Y\mathbb{P}^n$ of $\mathbb{P}^n$ along $Y$, and let $E$ be the exceptional divisor over $Y$. Then $E$ ...
3
votes
2answers
743 views

Ample divisors on blown-up projective space

Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let ...
1
vote
2answers
119 views

Divisors with positive Iitaka dimension

Let $X$ be a non-singular projective variety, and $D$ a divisor on $X$. Saying that $D$ has positive (meaning non-zero) Iitaka dimension is equivalent to the function $n \mapsto h^0(\cal{O}(D))$ ...
2
votes
4answers
343 views

Ample divisors on $\mathbb{P}^3$ blow-up along single point

Let $\pi:X\to\mathbb{P}^3$ be the blowing up at single point with $E$ be the exceptional divisor. Let $H=\pi^\ast\mathcal{O}_{\mathbb{P}^3}(1)$. In Ample divisors on the blow up of $\mathbb{P}^3$ at ...
3
votes
2answers
288 views

When is the Wendt binomial circulant determinant divisible by 3?

The Wendt binomial circulant determinant $W_n$ can be defined quite simply as a resultant: $$ W_n = \operatorname{res}(x^n-1, (x+1)^n-1). $$ Truer to its name, one may also define it as the ...
5
votes
1answer
215 views

Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?

Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$. We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective ...
0
votes
0answers
100 views

How to compute the Betti numbers of S-D for a surface S and a divisor D?

Let S be a projective non-singular surface and D a Cartier divisor which has a smooth representative. Can the Betti numbers of S-D be represented by the Betti numbers of S and D? In a paper ...
1
vote
0answers
182 views

Pull-back of globally generated sheaves

Let $X$ be a smooth projective surface in $\mathbb{P}^3$, $D=\sum_i n_iD_i$ an effective Cartier divisor. Let $C$ be a smooth irreducible curve on $X$. Denote by $i:C \hookrightarrow X$ is the closed ...
2
votes
1answer
399 views

On morphisms to projective space arising from a linear system

Context: This question arose as I was reading the proof of Application 6.1 in Mumford's Abelian Varieties. However, I have extracted all of the relevant information below so this question should ...
1
vote
1answer
119 views

divisors on $\overline{\mathcal{M}}_{g,n}$ that are trivial on certain $F$-curves

Inside the moduli space of curves $\overline{\mathcal{M}}_{g,n}$ one can distinguish two classes of $F$-curves isomorphic to $\mathbb{P}^1$: those of type $\overline{\mathcal{M}}_{0,4}$, and those of ...
1
vote
1answer
282 views

Does every ample divisor “span” a hyperplane?

Let $X\subset\mathbb{P}^n$ be a smooth projective variety of dimension $\geq 2$ and assume that it is not contained in any hyperplane. Now, take some hyperplane $H\subset\mathbb{P}^n$ and consider the ...
5
votes
1answer
277 views

Is there a canonical notion of principal divisor on a discrete dynamical system?

I hope this question is well-posed. Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian ...