All Questions
Tagged with divisors birational-geometry
98 questions
0
votes
1
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249
views
Behavior of divisors under push forward and pull back
Consider a birational morphism between smooth projective varieties $f:X\to Y$. I would like to understand the behavior of push-pull/pull-push of effective divisors under $f$. I know that if $D$ is an ...
- 101
2
votes
0
answers
244
views
On the definition of the relative canonical divisor
Fix a field $k$. Let $X,Y$ be normal integral $k$-schemes of finite type and $h: Y \to X$ a proper birational $k$-morphism. Moreover, assume that $X$ is Gorenstein, i.e. it admits a canonical divisor $...
- 293
16
votes
3
answers
4k
views
Contracting divisors to a point
This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up.
If $X$ is a projective variety over an algebraically closed field ...
- 2,305
2
votes
2
answers
208
views
Mori cones and projective morphisms
Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties, and $NE(X),NE(Y)$ the Mori cones of curves of $X$ and $Y$. Assume that $NE(X)$ is finitely generated. Then is $NE(Y)$ finitely ...
- 10.5k
0
votes
1
answer
257
views
Definition of canonical pair
Let $(X,D)$ be a pair and $f:Y\rightarrow X$ a log resolution. Write
$$
K_Y + \widetilde{D} = f^{*}(K_X) + \sum_{i}a_iE_i
$$
where $\widetilde{D}$ is the strict transform of $D$. I found the following ...
- 1,306
12
votes
1
answer
8k
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 $x_1\cdot...\...
- 151
2
votes
1
answer
171
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 $...
CommunityBot
- 1
4
votes
0
answers
1k
views
Is there any relation between the pushforward of a divisor and the pushforward of its line bundle?
Given a morphism between normal varieties $f: X \to Y$, we can push forward a Cartier divisor $D$ to get a cycle $f_* D$. On the other hand, we can form the line bundle $\mathscr{L}(D)$ and push that ...
- 4,164
4
votes
1
answer
230
views
Volume of conic bundles
Consider a smooth conic bundle $X\rightarrow \mathbb{P}^1$ with discriminant of degree $d$ (the locus of $\mathbb{P}^1$ over which the fibers are reducible conics). There is a formula for $(-K_X)^2$ ...
- 6,995
2
votes
1
answer
218
views
Existence of terminal $3$-fold flips
Does there exist a terminal $3$-fold $X$ with a curve $C\subset X$ such that $K_X\cdot C < 0$ admitting a Mori flip $X\dashrightarrow Y$, flipping $C$ to a curve $C'\subset Y$, where the singular ...
- 1,306
1
vote
0
answers
202
views
Mori cone of Picard rank two varieties
Let $X$ be a smooth projective variety of Picard rank two. Assume that there exists a surface $S\subset X$ such that
$$i^{*}:\text{Pic}(X)\rightarrow\text{Pic}(S)$$
is an isomorphism, where $i:S\...
- 549
3
votes
2
answers
968
views
Rationality of conic bundles
Let $\pi:X\rightarrow\mathbb{P}^2$ be a $3$-fold conic bundle, and let $\Delta\subset\mathbb{P}^2$ be its discriminant. Assume that both $X$ and $\Delta$ are smooth and that $deg(\Delta)\geq 6$.
Can ...
- 8,998
2
votes
1
answer
482
views
Divisors on projective bundles
Let $\pi:X = \mathbb{P}(\mathcal{E})\rightarrow\mathbb{P}^n$ be a projective bundle, where $\mathcal{E}$ is a rank two vector bundle over $\mathbb{P}^n$.
If $n = 0$ then $X = \mathbb{P}^1$, and for $n ...
- 431
5
votes
2
answers
527
views
Divisors whose restriction is big
Let $f:X\rightarrow Y$ be a flat morphism of smooth projective varieties, and $\mathcal{L}$ an effective and ample line bundle on $Y$. For a divisor $A\in H^0(Y,\mathcal{L})$ set $X_A := f^{-1}(A)$.
...
- 8,998
0
votes
1
answer
184
views
Curves in conic bundles
Consider a smooth minimal $3$-fold conic bundle $f:X\rightarrow\mathbb{P}^2$. Then $X$ has Picard rank two and consequently also the vector space of $1$-cycles is $2$-dimensional. Then the cone of ...
- 39.3k
6
votes
2
answers
524
views
Global sections of multiples of a divisor
Let $D$ be an integral divisor on a smooth projective variety $X$. Consider the multiples $mD$ of $D$ for $m\geq 0$. Clearly, $h^0(X,mD) = 1$ for $m = 0$.
Is there any example where $h^0(X,mD) = 0$ ...
- 149k
2
votes
1
answer
487
views
A question on effective divisors
Let $X$ be a projective variety with two morphisms $f:X\rightarrow Y$ and $g:X\rightarrow Z$ with irreducible fibers of positive dimension. Assume that $Pic(X) = f^{*}Pic(Y)\oplus g^{*}Pic(Z)$. Then ...
- 4,111
2
votes
1
answer
489
views
Nef and pseudo-effective divisors over non algebraically closed fields
Let $X$ be a projective variety over a field $K$. As a consequence of Kleiman's criterion, when $K$ is algebraically closed, we have that if $D$ is a nef divisor on $X$ then $D$ is pseudo-effective.
...
- 1,823
0
votes
1
answer
271
views
Pseudoeffective divisors on surfaces
Consider a minimal smooth conic bundle $S$ of dimension two. Assume that there are two curves $C,F$ on $S$ such that $C^2 < 0$ and $F^2 = 0$. Let $D$ be a pseudoeffective divisor on $S$ such that $...
- 42.9k
3
votes
1
answer
529
views
Examples of complex manifolds with trivial Néron–Severi group?
$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$Let $X$ be a compact complex manifold, assume projective if you'd like. Define the Néron–Severi group to be the quotient $$\NS(X) = \Pic(X) / \...
- 498
3
votes
1
answer
461
views
Pullback of $\mathbb{R}$-Cartier divisors
I am reading the recent book by Kawamata, Algebraic Varieties: Minimal Models and Finite Generation. There is an English translation here .
In the bottom of page 16 he says that an $\mathbb{R}$-...
- 1,307
9
votes
1
answer
443
views
Set theoretic equation for Veronese varieties
Consider the embedding $f:\mathbb{P}^n\rightarrow\mathbb{P}^N$ induced by the complete linear system of degree $d$ hypersurfaces of $\mathbb{P}^n$. Its image $V_{n,\,d}$ is degree $d$ Veronese variety ...
- 5,359
10
votes
2
answers
1k
views
Picard group of a cubic hypersurface
Consider the following cubic hypersurface in $\mathbb{P}^5$:
$$
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
$$
The singular locus of $X$ is the Veronese surface $V\...
- 39.3k
2
votes
0
answers
220
views
Divisorial contraction to a non-normal variety
Consider a divisorial contraction $f:X\rightarrow Y$, between projective varieties, contracting an irreducible divisor $D\subset X$ to a subvariety $Z\subset Y$ of codimension at least two, and which ...
- 63.9k
3
votes
1
answer
217
views
Weak Fano varieties and small transformations
A projective normal and $\mathbb{Q}$-factorial variety $X$ is said to be log Fano if there exists and effective divisor $D$ on $X$ such $-K_X-D$ is ample and the pair $(X,D)$ is klt.
Now, let $f:X\...
- 8,998
1
vote
1
answer
729
views
Push-forward of divisors and intersections
Let $f:X\rightarrow Y$ be a surjective finite morphism of varieties, with $X$ normal and $Y$ smooth. Let $D\subset X$ be a divisor and $C\subset Y$ a curve. Does the equality
$$C\cdot f_{*}D = f^{*}C\...
- 1,104
3
votes
0
answers
199
views
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
- 1
5
votes
0
answers
171
views
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
3
votes
0
answers
135
views
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
1
vote
0
answers
90
views
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
1
vote
1
answer
215
views
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
2
votes
0
answers
164
views
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
- 1
2
votes
0
answers
163
views
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
- 1
1
vote
0
answers
116
views
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
8
votes
0
answers
343
views
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 ...
- 227
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 $l=h-e$...
- 2,444
7
votes
1
answer
1k
views
Pull-back divisor being Cartier
Let $\pi \colon X \rightarrow Y$ be a projective morphism with connected fibers between normal quasi-projective varieties. Let $N$ be a $\mathbb{Q}$-Cartier divisor on $Y$ so that $\pi^*(N)$ is ...
- 4,111
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 ...
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 ...
CommunityBot
- 1
5
votes
0
answers
540
views
When does a Cartier divisor a pull-back of a Cartier divisor?
Suppose $f: Y \to X$ is a projective birational morphism between two varieties with mild singularities. For example, we can assume $X$ is normal and kawamata log terminal, $Y$ is $\mathbb Q$-factorial....
- 3,472
3
votes
1
answer
295
views
Infinitely small intersections with nef $\mathbb R$-Cartier divisors
Suppose $X/\mathbb C$ is a projective $\mathbb Q$-factorial variety with wild singularities. Let $N$ be a nef $\mathbb R$-Cartier divisor. Then is it possible that there are infinitely many curves $...
- 28.8k
3
votes
1
answer
292
views
Ring of sections and normalization
Let $D$ be a base-point-free divisor on a normal projective variety $X$, and let $Y$ be the image of the morphism $f_{D}:X\rightarrow Y$ induced by $D$. Assume that $f_D$ is birational.
Now, let $X(D)...
- 4,111
2
votes
1
answer
257
views
Flipping and flipped loci
Let $f:X\dashrightarrow Y$ be the flip of a small contraction $\phi:X\rightarrow Z$, and let $\psi:Y\rightarrow Z$ be the small contraction such that $\psi\circ f = \phi$. Let $Exc(\phi), Exc(\psi)$ ...
- 1,164
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 ...
- 1,164
14
votes
1
answer
529
views
Birational automorphisms of varieties of Picard number one
Let $X$ be a smooth projective variety of Picard number one, and let $f:X\dashrightarrow X$ be a birational automorphism which is not an automorphism.
Must $f$ necessarily contract a divisor?
- 223
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 ...
- 7,249
6
votes
2
answers
719
views
Intersection numbers in $\mathbb{P}^1$-bundles
Let $\mathcal{E}$ be a rank two vector bundle on $\mathbb{P}^2$ fitting in the following exact sequence
$$0\rightarrow \mathcal{O}_{\mathbb{P}^2}\rightarrow \mathcal{E}\rightarrow \mathcal{I}_p(-1)\...
CommunityBot
- 1
7
votes
1
answer
945
views
Push-forward of nef divisors via finite morphisms
Let $f:X\rightarrow Y$ be a finite morphism between smooth projective varieties, and let $D$ be an effective nef but not ample divisor on $X$.
Consider the divisor $f_{*}D$ on $Y$. Is $f_{*}D$ nef ...
- 38k
5
votes
1
answer
585
views
Anti-canonical divisor of a Fano variety
Let $X$ be a normal projective Fano variety, that is the anti-canonical divisor $-K_X$ is ample.
For any $m>0$ let us consider the complete linear system $|-mK_X|$ and the map
$$f_{|-mK_X|}:X\...
- 1,164
3
votes
2
answers
624
views
Is this divisorial contraction a blow-up?
Let $C$ be a curve in a smooth $3$-fold $X$ with an ordinary node $p\in X$. Blow-up $p$ let $E$ be the exceptional divisor, and $\widetilde{C}$ the strict transform of $C$. Furthermore let $L$ be the ...
- 8,998