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.
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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 ...
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birational geometry of moduli spaces: why work on the coarse space?
In studying the birational geometry of $\overline{\mathcal{M}}_g$, it seems standard to work with the coarse space $\overline{M}_g$ rather than the smooth stack $\overline{\mathcal{M}}_g$. Why is this?...
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Curves on rational surfaces and Lang's conjecture for M_g
There are a group of related conjectures associated to Lang's name - for this question I'll consider only the weakest one, namely that rational curves in a projective variety of general type are not ...
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Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?
The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...
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References about conic bundles
I'm interested in stable rationality of conic bundles by means of Brauer group/unramified cohomology non-triviality, and I was wondering if there are some references for the basic properties of conic ...
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Letters of a bi-rationalist
V.V. Shokurov has written several papers over the course of about 10 years which are called "Letters of a bi-rationalist". Here are the ones that I could find:
Letters of a bi-rationalist. I. A ...
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How can I "see" that a map is birational?
This came up with the Euler brick.
Let $T=(p,q,r)$ be a Randall triple, i.e. $$(p^2-1)(q^2-1)(r^2-1)=8pqr\ \qquad\text{[eq.1]}.$$ There are tons of maps that map a triple $T$ to another $T'=(p',q',r')$...
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How many characteristics is a random surface unirational in?
Suppose I have a surface $X$ defined over $\mathbb{Z}$. I am interested in the set $S_X$ of primes $p$ such that $X_{\overline{\mathbb{F}}_p}$ is unirational. If I choose a "random" surface ...
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Which field extensions do not affect Chow groups?
Let $X$ be a (say, smooth projective) variety over a field $k$. For which $K$ it is known that the ("ordinary", that is, not higher) Chow groups of $X$ map onto that of $X_K$ bijectively?
...
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On a paper of Formanek about $PGL_4$
In his paper "The center of the ring of 4 × 4 generic matrices" (Journal of Algebra, 1980) Formanek shows that, if $V$ is the representation of $PGL_4$ given by simultaneous conjugation of pairs of ...
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Motivic homotopy theory and Noether problem
Let $G$ be a finite group, and let $V$ be a faithful representation of $G$. The Noether problem asks whether $V/G$ is rational (stably rational, retract rational) or not.
To construct ...
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Rationality of a certain real algebraic variety
Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even.
Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...
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Maximally nodal degree 6 Fano threefolds
Let $X$ be a complete intersection of a quadric and a cubic in $\mathbb{P}^5$. In the smooth case it is a so-called Fano threefold of index one and degree six.
I would like to consider the case when $...
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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 ...
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Status of the Sarkisov program a la Corti
To a birational map $f$ between Mori fiber spaces, one can associate the "Sarkisov degree", a triple $(\mu,\lambda,e)$ which I will not define here. To factor $f$ into a sequence of elementary ...
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Number of rational points over finite fields mod $q$ is birational invariant
I heard that if $\mathbf F_q$ is a finite field, $X, Y$ are birational smooth proper variety over $\mathbf F_q$, then $\#(X(\mathbf F_q)) \equiv \#(Y(\mathbf F_q)) \pmod q$, and I heard that the proof ...
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Log Calabi-Yau variety diffeomorphic to an algebraic torus
Let $U$ be a complex affine log Calabi-Yau variety, which I take to mean a smooth affine variety which admits a compactification in a smooth projective variety $X$ with an snc anti-canonical ...
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Projectivity of flops
Say I have a projective (smooth, compact) irreducible symplectic variety $X$ over $\mathbb{C}$ and I perform a Mukai flop. It is well known that if the resulting variety $\widetilde{X}$ is Kahler, it ...
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Different definition of Cox rings
Definition: Let $X$ be a normal projective variety with finitely generated Picard group. Define the Cox ring of $X$ as the multisection ring $$\text{Cox}(X)=\bigoplus_{(m_1,\ldots,m_k)\in \mathbb{N}^k}...
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Conic bundles which are rational but with non-rational generic fibres
Let $n\ge 1$ be an integer and let us work over the field of complex numbers. Let $\mathcal{R}_n$ denote the set of rational conic bundles $\pi\colon X\to \mathbb{P}^n$ (morphisms such that the ...
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"Reflexive" differentials on Gorenstein affine toric variety
Let $P \subset \mathbb{R}^{n-1}$ be a lattice polytope of dimension $n-1$ and let $\sigma \subset \mathbb{R} \times \mathbb{R}^{n-1}$ be the cone over $1 \times P$.
To the cone $\sigma$, we may ...
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Pseudo-effective divisor which is not nef in any birational model
Let $X$ be a smooth complex projective algebraic variety and let $D$ be a $\mathbb{Q}$-Cartier pseudo-effective divisor on $X$. Lets say that $D$ is birationally nef
if there exists a birational ...
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Restriction of the Canonical Divisor $K_X$ to a general fiber
Let $\ f:X\to Z$ be a surjective morphism between two smooth projective varieties with connected fibers $(f_*\mathcal{O}_X=\mathcal{O}_X)$. Let $F$ be a general fiber of $f$ and $\mbox{dim } F<(\...
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Equations for conic del Pezzo surfaces of degree one
Let $X$ be a del Pezzo surface of degree one over a field $k$ of characteristic not $2$ equipped with a conic bundle $\pi: X \rightarrow \mathbb{P}^1$. By Theorem 5.6 of this paper, $X$ admits a ...
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Strong factorisation conjecture for toric varieties
In this survey is remarked (see page 6 after Example 1.12) that to prove the
Conjecture 1.10 (Strong factorisation). Let $\phi: X \dashrightarrow Y $ be a birational map
between two quasi-projective ...
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The structure of the Hilbert scheme of conics contained in hypersurfaces in $\mathbb P^3$
We work over a field of characteristic $0$. Let $X\hookrightarrow\mathbb P^3$ be a geometrically integral hypersurface of degree $\delta$. It is well known that the Hilbert scheme of conics in $\...
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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 ...
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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 ...
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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 ...
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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 = \sum_i a_i D_i$ be a $\mathbb{Q}$ divisor. Assume ...
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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....
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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 ...
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Description of flop as graded algebra
I am looking for an example of a flop $Y \to X \leftarrow W$, possibly with exceptional locus at least a $\mathbb{P}^2$, where $X = \text{Spec } A$ is affine and $Y,W$ can be described as explicit ...
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What is the fundamental group of Kontsevich's space of stable maps?
... at least in the case where the target is a rationally connected variety.
This question is a follow-up to question
Constructing embedded families of curves with general moduli
and Jason Starr's ...
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Semistable sheaves on rationally connected manifolds
Let $(X,H)$ be a polarized projective manifold of dimension $n$ defined over $\mathbb C$, and let $\mathcal E$ be a reflexive sheaf on $X$.
If for every subsheaf $\mathcal F \subset \mathcal E$ the ...
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Existence of a rational curve in the center of a birational contraction for symplectic singularities
Let $M$ be a holomorphically symplectic
complex manifold, and $f: M \to X$
a holomorphic, birational contraction to a Stein
variety $X$, contracting a subvariety $E$
to a point, and bijective outside ...
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Do rational maps to abelian varieties extend across rational singularities?
Let $X$ be a normal proper variety with only rational singularities and $A$ an abelian variety. Does a rational map $X \supset U \to A$ extend to a morphism $X \to A$? If not, what is a ...
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Bondal-Orlov conjecture on Calabi-Yau varieties
Recently, I am trying to study the various progress made on the Bondal-Orlov conjecture: Birational Calabi-Yau varieties ⟹ Equivalent derived categories.
I have started reading the paper by Bridgeland ...
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Models of conic bundles
Let $S$ be a smooth projective variety over $\mathbb{C}$. A conic bundle over $S$ is a smooth projective variety $X$ together with a flat morphism $\pi:X \to S$ all of whose fibres are isomorphic to ...
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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 ...
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Parameter spaces for conic bundles
A conic bundle over $\mathbb{P}^n$ is a morphism $\pi:X\rightarrow\mathbb{P}^n$ with fibers isomorphic to plane conics. A conic bundle $\pi:X\rightarrow\mathbb{P}^n$ is minimal if it has relative ...
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Minimal $b_2$ in Sarkisov's construction
In the paper On the structure of conic bundles. Math. USSR, Izv.,
120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}...
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Dominant rational maps and compositions
According to many books (and also to Stack project, see https://stacks.math.columbia.edu/tag/01RI ), a morphism $f\colon X\to Y$ between schemes is said to be dominant if the image is dense in $Y$.
...
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When is the dualizing sheaf globally generated?
Let $X$ be a projective integral Cohen-Macaulay variety (over $\mathbb{C}$, if that makes things easier). The Cohen-Macaulay condition says that the dualizing complex (see this answer) is concentrated ...
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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 ...
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Calculate Kodaira dimension of a singular hypersurface
For a smooth projective hypersurface $H \subseteq \mathbb{P}^n$ of degree $d$ one can calculate its Kodaira dimension $\kappa(H)$, and find
$$\kappa(H) =
\begin{cases}
-\infty \qquad &\mbox{if } ...
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Behavior of geometric fibers under strict transforms
Consider a digram of the form
$$
\begin{array}{ccc}
Z' & \rightarrow & Z \\
\downarrow & & \downarrow \\
X' & \rightarrow & X \\
\downarrow & & \downarrow \\
S' &...
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Complete intersections in projective spaces
Let $X$ be an arbitrary smooth projective variety over a field $k$.
Do there exist:
a smooth complete intersection $X'$ in a projective space.
a surjective morphism of $k$-varieties $X'\to X$ ?
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Pullback of the canonical bundle
Let $f : X\to Y$ be a morphism of smooth projective varieties over a field $k$.
Assume $f^*\omega_{Y/k} \simeq \omega_{X/k}$.
I'd like to collect a bestiary of the properties $f$ has, or even ...
4
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answers
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Singularities of Viehweg's fiber product
Let $f:X\to Y$ be a fibration between smooth projective manifolds. Assume that $\Delta$ is a simple normal crossing divisor in $Y$ such that $f^*(\Delta)$ is normal crossing in $X$, and
$$
f_0:X_0=f^{...