Questions tagged [birational-geometry]

Birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.

Filter by
Sorted by
Tagged with
1
vote
0answers
104 views

Realising flips as quotients of flops by an involution

In Séminaire BOURBAKI, 4ième année, 1988-89, n° 712, Juin 1989, Kóllar wrote the beautiful article Minimal Models of Algebraic Threefolds: Mori's program. Firstly, I would highly encourage this ...
2
votes
0answers
87 views

Determining if a morphism is a blowup along a given subvariety

Let $X,\tilde{X}$ be two smooth projective varieties over $\mathbb{C}$, and let $\pi:\tilde{X}\rightarrow X$ be a projective morphism. Let us moreover assume that there exists a smooth closed ...
2
votes
0answers
98 views

Certain endomorphisms of $\mathbb{C}(x,y)$

Let $f: (x,y) \mapsto (p,q)$ be a $\mathbb{C}$-algebra endomorphism of $\mathbb{C}(x,y)$ satisfying the following two conditions: (i) $\operatorname{Jac}(p,q):=p_xq_y-p_yq_x \in \mathbb{C}-\{0\}$. (...
6
votes
0answers
102 views

Picard group of resolution

Let $X$ be a normal variety and $f:Y\rightarrow X$ a birational morphism, contracting exceptional divisors $E_1,\dots,E_k$ onto the singular locus of $X$, with $Y$ smooth. In this situation is $Pic(Y)...
6
votes
0answers
113 views

Lefschetz type theorems for linear sections

Let $X\subset\mathbb{P}^n$ be e normal variety, $L\subset\mathbb{P}^n$ a linear subspace, and $Y = X\cap L$ a linear section. Assume that $Y$ is also normal. In particular, we have that $Sing(X)$ has ...
9
votes
1answer
195 views

A property of varieties between unirational and retract rational

EDIT: The vague question Q1 below is partially answered, while the concrete question Q2 seems to be still open. Let $V$ be a geometrically integral variety over a field $K$. I consider the following ...
1
vote
0answers
67 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
vote
0answers
101 views

Does nefness carry over through flips?

Suppose $\pi: X \dashrightarrow Y$ is a birational map which is an isomorphism in codimension 1 (such as a flip). Also suppose both $X$ and $Y$ have reasonable (say log terminal) singularities. We ...
3
votes
0answers
131 views

What is the exceptional divisor of a divisorial contraction?

Let $\pi \colon X\to Y$ be a divisorial contraction between $\mathbb{Q}$-factorial terminal varieties. Let $E\subset X$ be the exceptional divisor and let $\pi(E)=D$ be its image. It is true that $\...
1
vote
1answer
83 views

Mori extremal contraction with small Betti number

Is there example of a smooth, projective, complex $3$-fold $X$, having $b_{2}(X)=2$ a Mori extremal contraction $\phi: X \rightarrow X'$ which contracts a smooth quadric surface $Q \subset X$? It ...
2
votes
1answer
95 views

Cremona transformation $\sigma: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2 $ and pushforward of divisor

Let consider the birational Cremona transformation $\sigma: \mathbb{P}^2 \dashrightarrow \mathbb{P}^2 $ defined by $$(X:Y:Z) \mapsto (X^{-1}:X^{-1}: Z^{-1})=(YZ:XZ:XY)$$. In language of birational ...
2
votes
1answer
164 views

Picard group modulo codimension 2

Let $X$ be a normal (possibly singular) projective surface over $\mathbb{C}$. Consider the set $M_X$ of all coherent sheaves $F$ on $X$ such that there exists a finite subset $Y\subset X$ such that $F$...
1
vote
1answer
106 views

Is $\mathbb{Q}$-factoriality preserved under contraction?

Let ($X$,$\Delta$) be projective klt pair and $f \colon X \rightarrow Z$ be contraction of ($K_X + \Delta$) - negative extremal ray $R$. If $X$ is $\mathbb{Q}$ -factorial and $\mathrm{dim}Z < \...
4
votes
0answers
99 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:...
5
votes
0answers
113 views

Jacobian fibration of an abelian fibration

Let $f \colon S \rightarrow C$ be a minimal elliptic surface and let $g \colon J \rightarrow C$ be its jacobian fibration. In this case, we know that the fibers of $g$ are better behaved that the ones ...
4
votes
1answer
97 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 $p\in X$....
1
vote
0answers
70 views

Concerning $\mathbb{C}(s_1,s_2,s_3,y)=\mathbb{C}(x,y)$, where $s_1,s_2,s_3$ are symmetric

Perhaps the following question is not in the level of MO questions, but it has not received comments in MSE, so I ask it here also: Let $\beta: \mathbb{C}[x,y] \to \mathbb{C}[x,y]$ be the involution ...
1
vote
1answer
136 views

On the birational equivalent class of algebraic surfaces with Picard number $1$

An open subset $U$ of a projective surface $Z$ is big if $\mathrm{codim}_Z(Z\setminus U)\geq2$. Let $X$ and $Y$ be smooth complex projective surface. If there exists a birational map $f:X\...
1
vote
0answers
79 views

Characterizing subfields $\mathbb{C}(u,v) \subseteq \mathbb{C}(x,y)$ invariant under an involution

Let $\iota$ be an involution on $\mathbb{C}(x,y)$, namely, a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$ of order two. Examples of involutions: $\alpha: (x,y) \mapsto (y,x)$, $\beta: (x,y) ...
9
votes
1answer
218 views

Concerning $k \subset L \subset k(x,y)$

The following is a known result in algebraic geometry: Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$). Let $L$ be a field such that $k \subset L \subset ...
1
vote
0answers
93 views

A formula on a generically finite morphism

In Nakayama's book 2004, pg. 39, a formula is written: Let $f:X \to Y$ be a generically finite and proper surjective morphism, $D$ be a Cartier divisor on $Y$. Then, $f_*f^*D=(\deg f) D$. More ...
1
vote
0answers
60 views

Connected components of a codimension one fiber for a finite morphism

Let $f:X \to Y$ be a finite surjective morphism from a $\mathbb{Q}$-factorial variety to a smooth variety. Let $D_Y$ be a prime divisor on $X$ and let $\bigcup D_i$ be the inverse image of $D_Y$. Do ...
3
votes
1answer
157 views

Kodaira dimensions of push-forward via finite map

Let $f:X \to Y$ be a finite map from a normal projective variety to a smooth projective variety, $D$ be a Cartier divisor on $X$. Do we have any relation between $\kappa(X,D)$ and $\kappa(Y,f_*D)$?
3
votes
1answer
219 views

What is known about lower etale cohomology of unirational varieties?

Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite ...
2
votes
0answers
75 views

Log canonical centers of toric (and toroidal) varieties

Q1: Let $(X,B)$ be a toric variety. There exists a toric resolution of singularities $f:(Y,E) \to (X,B)$. Here is my question: Is any lc center of $(X,B)$ an irreducible component of an intersection ...
1
vote
0answers
67 views

Birational model of a log smooth pair

Given a log smooth pair $(X,B)$ with a reduced boundary divisor $B$, consider a birational model $\pi:X' \to X$ and a boundary divisor $B'$ which is given by $K_{X'}+B'=\pi^*(K_X+B)$. Here is my ...
4
votes
0answers
95 views

Are there any explicit (prime-to-l) alterations for interesting varieties (or schemes)?

I have read that it is easier to find regular alterations of varieties than their resolutions of singularities (moreover, I believed in this sentence when I read it). My question is: do there exist ...
1
vote
0answers
101 views

Are terminal singularities $ \mathbb{Q}$-factorial?

The proof of Lemma 5-1-5 in this 1987 paper by Kawamata, Matsuda and Matsuki (link on Projecteuclid) seems to say that a variety with terminal singularities is $\mathbb{Q}$-factorial ( I only need ...
2
votes
0answers
60 views

Finding divisors with canonical singularities in a moving linear system

I apologize if the question is too naive or trivial: We know that any reduced divisor in a smooth variety has Gorenstein singularities. However, I don't know if there's a cone theorem for Gorenstein ...
2
votes
0answers
66 views

Birational contraction of toric vector bundle

Let $X$ be the toric vector bundle over $\mathbb{P}(1,1,1,2)$ with grading matrix $$ \left(\begin{array}{cccccc} 1 & 1 & 1 & 2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 & ...
3
votes
0answers
131 views

Explicit equations for rational elliptic surfaces (Halphen surfaces)

I am looking for explicit equations for rational elliptic surfaces in characteristic $2$. For me, a rational elliptic surface $X$ is a smooth projective surface $X$ which is rational and equipped with ...
2
votes
1answer
78 views

Strict transform by normalization

I am trying to understand the following definition in C. Sabbah's paper (Quelques remarques sur la géométrie des espaces conormaux), page 186, Numdam link. Let $\phi\colon X\to \mathbb{C}^2$ be a map ...
5
votes
0answers
105 views

Unirationality of universal Jacobian over special strata of moduli space of pointed genus 3 curves

Let $M_{3,1}$ be the (coarse, non-compactified) moduli space of genus $3$ curves with a marked point over a field $k$ of characteristic zero. Throwing away the hyperelliptic curves, take the open ...
1
vote
1answer
205 views

Some simple algebra of rational functions by André Weil

In André Weil's dissertation, he considers two meromorphic functions $x,y$ on a complex curve.  He assumes every pole of $y$ is a pole of $x$, and its multiplicity as a pole of $y$  is no greater than ...
3
votes
0answers
100 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 $X$ ...
1
vote
0answers
68 views

Has anyone researched this variant of separable rational connectedness?

If I am correct, then one of the definitions of rational chain connectedness is that a variety $ X $ is rationally chain connected if 1) there are schemes $ \mathcal{C} $ and $ T $ with a flat, ...
1
vote
1answer
138 views

Does anyone know a reference in the literature regrading a proof that every projective hypersurface with vanishing canonical divisor is uniruled

I saw a result in notes on by Olivier Debarre (Rational Curves on Hypersurfaces, Lecture notes for the II Latin American School of Algebraic Geometry and Applications 1-12 of June 2015 in Cabo Frio, ...
0
votes
0answers
185 views

Does anyone know of a uniruled hypersurface over $ \mathbb{C} $, which is not rationally connected?

Does anyone know of any projective, hypersurfaces $ Z $ which are uniruled, but not rationally connected such that $ Z $ is not an $ \mathbb{A}^{1}_{\mathbb{C}} $ bundle over another variety. Note ...
8
votes
2answers
233 views

Finiteness of birational types for targets of algebraic fibrations

Let $X$ be a smooth projective variety. A fibration is a surjective map with connected fibers between projective varieties. Is it true that there's a finite number of birational equivalence classes of ...
3
votes
1answer
264 views

Are Gromov-Witten invariants birational invariants?

Let $X$ and $Y$ be two smooth complex projective varieties. Then they are also symplectic manifolds. We know Gromov-Witten (GW) invariants are symplectic invariants. That means if there exists a a ...
1
vote
0answers
68 views

Is the boundary divisor of a smooth projective toric variety an snc divisor?

Let $X$ be a smooth toric projective variety. Let $T$ be the big torus acting on $X$. Let $D=X\backslash T$ be the boundary divisor. Question 1. Will $D_i$ be a smooth toric projective variety for ...
3
votes
1answer
248 views

Push forward a Cartier is still Cartier

Suppose that $X$ is a projective complex variety with rational singularities (i.e. for any resolution $f: Y \to X$, $f_* \mathcal{O}_Y=\mathcal{O}_X, Rf^i_*\mathcal{O}_Y=0~ \forall i \geq 1$). Let $f: ...
3
votes
2answers
238 views

When is a monomial rational map on the projective space birational?

Let $k$ be an algebraically closed field of characteristic $0$. For $\alpha :=(a_1,\dots,a_{n+1})\in \mathbb N^{n+1}_{\ge 0}$ , let $\bar x^{\alpha}:= x_1^{a_1} \dots x_{n+1}^{a_{n+1}} \in k[x_1,\...
2
votes
0answers
149 views

definition of discrepancy

In Kollar and Mori "Birational geometry of algebraic varieties" discrepancy is defined as following way. Let X be a normal variety and $D = \Sigma_i a_i D_i$ be a $\mathbb{Q}$ divisor . Assume that $...
1
vote
2answers
201 views

Flawed argument and when is a sheaf that can be associated to any complete, normal variety a birational invariant?

Theorem 8.19 of Hartshorne states the following: Let $ X $ and $ X^{'} $ be two birationally equivalent nonsingular projective varieties over $ k $. Then $ p_{g}(X) = p_{g}(X^{'}) $. I thought of ...
1
vote
2answers
211 views

$\operatorname{NEF}(X)\subset\operatorname{Big}(X)$?

Does there exist a smooth projective variety $X$ such that $\operatorname{NEF}(X)-\{0\}$ is strictly contained in $\operatorname{Big}(X)$, where $\operatorname{Big}(X)$ is the interior of $\overline{\...
3
votes
0answers
103 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?
1
vote
0answers
184 views

Motivic integration of an Abelian variety and its dual are same?

Let $A$ be an abelian variety over $\mathbb{C}$ and $A^*$ the dual Abelian variety. The class of $A$ and the class of $A^*$ in $\mathcal{M}_{\mathbb{C}}=K_0(Var_\mathbb{C})[\mathcal{L}^{- 1}]$ are ...
2
votes
0answers
73 views

Integrality of divisors in the canonical bundle formula

Suppose $f: X \to Z$ is a fibration with connected fibers between normal projective varieties. Suppose $(X, \Delta)$ is log canonical and $K_X+\Delta\sim_\mathbb{Q} 0/Z$, then there is a canonical ...
2
votes
0answers
90 views

The Weil restriction of an elliptic curve with respect to $\mathbb{F}_{p^2}/\mathbb{F}_{p}$

For a prime $p > 3$ consider the quadratic finite field extension $\mathbb{F}_{p^2}/\mathbb{F}_{p}$. Also, consider the elliptic curves $$ E\!: y_0^2 = x_0^3 + ax_0 + b,\qquad E^{(1)}\!: y_1^2 = ...

1
2 3 4 5
10