Trending 'ag.algebraic-geometry+birational-geometry+divisors' questions - Page 2

6 votes

1 answer

1k views

Picard groups and birational morphisms

Let $f:X\rightarrow Y$ be a birational morphism of projective varieties. Assume that $Pic(X)$ is a free abelian group generated by $n$ divisors $D_1,...,D_n$. Under which hypothesis on $X$ and $Y$ is ...

user97096

asked Nov 21, 2016 at 15:29

5 votes

1 answer

244 views

Blowing-up an ideal generated by squares

Let $f_1,\dots,f_r$ be regular functions on a smooth projective variety $X$, and consider the ideals $I = (f_1^2,\dots,f_r^2)$ and $J = (f_1,\dots,f_r)$. Let $Y = Z(I)$ and $W = Z(J)$ be the ...

user114666

asked Nov 9, 2017 at 0:22

2 votes

1 answer

209 views

Curves contracted by a rational map

Let $D$ be a big but not nef divisor on a normal $\mathbb{Q}$-factorial projective variety. Assume that the section ring $$R(D) = \bigoplus_{n\in\mathbb{N}}H^0(X,nD)$$ is finitely generated and ...

user114666

asked Dec 16, 2017 at 17:42

5 votes

2 answers

656 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 ...

user58018

asked Apr 12, 2015 at 18:34

2 votes

1 answer

974 views

Inverse image of a divisor

Let $f:X\rightarrow Y$ be a morphism with connected fibers between projective varieties (not necessarily flat). Let $D\subset Y$ be an irreducible divisor. Let us look at the cycle $f^{-1}(D)\subset X$...

user82886

asked Feb 23, 2016 at 16:53

2 votes

1 answer

511 views

Rigid effective divisors

Let $D\subset X$ be an effective smooth divisor in a smooth projective variety $X$. Assume that $h^0(X,D)=1$. In particular $D$ spans an extremal ray of the effective cone of $X$. Now, let $f:X\...

user77919

asked Feb 11, 2016 at 21:17

2 votes

1 answer

679 views

A Decomposition for Iitaka fibration

Let $\pi: X\to Y$ be an Iitaka fibration of projective varieties $X,Y$, then is there always the following decomposition $$K_Y+\frac{1}{m!}\pi_*\mathcal O_X(m!K_{X/Y})=P+N$$ where $P$ is ...

user21574

asked Aug 2, 2016 at 1:25

3 votes

2 answers

5k 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 ...

user47036

asked May 15, 2014 at 20:26

3 votes

1 answer

453 views

Extremal rays of the effective cone

Let $X$ be a smooth projective variety with polyhedral finitely generated effective cone $Eff(X)$. Let $f:X\dashrightarrow X$ be a birational automorphism of $X$ that is an isomorphism in codimension ...

user56259

asked Nov 17, 2016 at 18:51

6 votes

1 answer

3k 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 $E$ ...

user49214

asked Apr 8, 2014 at 20:59

2 votes

1 answer

181 views

Anti-canonical divisorial contractions of weak Fano $3$-folds

Let $X$ be a smooth weak Fano but not Fano $3$-fold ($-K_X$ is nef and big but not ample). Then the anti-canonical morphism $\phi:X\rightarrow W$ (the morphsim induced by the linear system $|-mK_X|$ ...

user114198

asked Sep 15, 2017 at 0:57

1 vote

1 answer

240 views

Divisor class group of quartic surfaces

Let $X\subset\mathbb{P}^3$ be a normal quartic surface with divisor class group $Cl(X)\cong\mathbb{Z}[H]$ generated by the hyperplane section. What can we say about the singularities of $X$?

user97096

asked Mar 1, 2017 at 21:39

3 votes

3 answers

732 views

Nefness of $h-e$ in the blowup of $\mathbb{P}^n$

Let $S$ be the blow up of $\mathbb{P}^n$ in a point $P$. Let $h$ be the class of the pullback of an hyperplane of $\mathbb{P}^n$ and $e$ the class of the exceptional divisor. Why is the divisor $l=h-e$...

Fulvio

31

asked Jan 29, 2010 at 17:43

3 votes

1 answer

264 views

Linear systems on moduli spaces of stable maps

I am studying the general theory of moduli spaces of stable maps, in particuar of the moduli spaces $\overline{M}_{0,n}(\mathbb{P}^r,d)$ of degree $d$ stable maps from a rational curve with $n$ marked ...

user97096

asked Jun 7, 2017 at 9:24

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

asked Feb 2, 2018 at 22:49

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

asked Mar 16, 2017 at 19:47

1 vote

1 answer

273 views

Big divisors in family

Given a family of divisors $D_t$ on varieties $X_t$, there are examples that show that bigness is not well behaved (e.g. example 2.2.13 in Positivity 1, shows we can have a special fiber where $D_0$ ...

Stefano

625

asked Nov 22, 2016 at 18:41

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 ...

Stefano

625

asked Jun 26, 2017 at 18:01

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 ...

Mohsen Karkheiran

503

asked Aug 6, 2018 at 3:31

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 $\...

user41650

1,982

asked Apr 5, 2017 at 9:44

6 votes

2 answers

483 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 $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$. ...

user58018

asked Dec 21, 2014 at 14:15

2 votes

2 answers

674 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 ...

user58018

asked Dec 12, 2014 at 10:33

1 vote

0 answers

312 views

Cone of moving curves

Let $X$ be a projective variety and $C\subset X$ be a moving curve, that is the curves numerically equivalent to $C$ cover a dense open subset of $X$. How can we detect when $C$ is an extremal ray ...

i87456

141

asked Jan 26, 2017 at 22:26

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_{...

Puzzled

8,998

asked Jul 8, 2017 at 14:54

1 vote

2 answers

404 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 particular,...

user68440

asked Feb 23, 2015 at 21:24

0 votes

1 answer

342 views

Intersections of divisors in blow-ups of $\mathbb{P}^n$

Let $p_1,p_2,p_3\in\mathbb{P}^n$ be three general points, $X$ the blow-up of $\mathbb{P}^n$ at $p_1,p_2,p_3$, then along the lines $\left\langle p_i,p_j\right\rangle$, and finally along the plane $\...

user91659

asked Sep 30, 2016 at 20:07

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 ...

Sabina

79

asked Jul 2, 2017 at 22:40

1 vote

1 answer

242 views

Number of conditions imposed by fat points to a linear system

Let $|D|$ be the linear system of degree $d$ hypersurfaces in $\mathbb{P}^n$ having multiplicity at least $m$ at $s$ general points. Then $|kD|$ is the linear system of degree $kd$ hypersurfaces in $...

user97096

asked Sep 23, 2016 at 22:30

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

asked May 24, 2016 at 11:11

1 vote

1 answer

503 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.

user58018

asked Feb 1, 2015 at 17:56

1 vote

1 answer

635 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 ...

user47036

asked Jul 5, 2014 at 2:12

3 votes

1 answer

125 views

Sections of a linear system splitting as a product of degree one polynomials

Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ and with multiplicities $m_1,...,m_k$ at $p_1,...,p_k\in\mathbb{P}^n$ general points. Let $S\subseteq |\mathcal{O}_{\mathbb{P}^n}(d)|$ be ...

user75933

asked Apr 28, 2016 at 14:25

0 votes

2 answers

490 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 $(X,\...

Puzzled

8,998

asked Jul 17, 2014 at 15:21

1 vote

0 answers

70 views

Moving curves and small transformations

Let $f:X\dashrightarrow Y$ be an isomorphism in codimension one between smooth projective varieties. Let $C\subset X$ a curve generating an extremal ray of the cone of moving curves $Mov_1(X)$, and ...

i87456

141

asked Jan 27, 2017 at 22:15

3 votes

1 answer

333 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 ...

user39380

asked May 26, 2014 at 14:31

0 votes

1 answer

259 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 $H$ ...

user39380

asked Jun 1, 2014 at 14:43

1 vote

1 answer

687 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 $mult_xD < 1$ ...

user58018

asked Sep 10, 2014 at 0:22

1 vote

1 answer

248 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 $...

user49214

asked May 26, 2014 at 16:34

7 votes

1 answer

650 views

Cones, monoids, and the space of (very) ample divisors

An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...

Noah Giansiracusa

1,871

asked Mar 31, 2011 at 12:45

2 votes

2 answers

1k views

Global sections of a linear system

Recently, following Beauville's book (exercises iv.(1),(2)) I have been working on Hirzebruch surfaces (from the algebraic geometry point of view) and I had to compute the space of global sections of ...

Jesus Martinez Garcia

2,215

asked Jul 12, 2011 at 11:31

0 votes

1 answer

248 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 ...

user49214

asked Jun 17, 2014 at 11:05

3 votes

1 answer

1k views

Birational pullbacks of divisors on singular varieties

Actually I have two related questions. Here is the first... Suppose $X$ is a, possibly singular, complex projective variety. Let $D$ be an effective Cartier divisor on $X$ and $x\in X$ a closed ...

Gianni Bello

1,150

asked Jun 7, 2010 at 20:05

2 votes

1 answer

389 views

resolution of singularities and a projection formula

Let $Y$ be a normal surface and let $p:X\longrightarrow Y$ be a resolution of singularities. Let $f$ be a rational function on $Y$. Do we have that $p_\ast$div $(d(f\circ p)) = $ div $df$ as cycles?...

Inhop

21

asked Jul 22, 2011 at 15:06

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 ...

Karl Schwede

20.5k

asked Oct 28, 2010 at 4:50

2 votes

1 answer

557 views

Numerically rigid nef divisor

Is it possible to find an example of an $\mathbb{R}$-Cartier divisor $D$ on an irreducible variety $X$ that is non-trivial, nef, effective and numerically rigid? By "numerically rigid" I mean that ...

Oren

205

asked Nov 16, 2010 at 11:27

2 votes

1 answer

269 views

Divisor intersecting non-negatively the negative part of its Zariski decomposition

Hi all. I'm looking for an example of a smooth projective surface $X$ and a pseudo-effective divisor $D$ on $X$ such that when I consider the Zariski decomposition $D=P+N$ there is some component $E$ ...

Gianni Bello

1,150

asked Nov 16, 2011 at 13:57

2 votes

1 answer

743 views

About b-divisors

In the last period I have studied a number of papers of O. Fujino and F. Ambro using the language of b-divisors. So far it seems to me that every proof I have studied can be translated in the ...

Gianni Bello

1,150

asked May 31, 2010 at 9:04

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