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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questions with no upvoted or accepted answers
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Is there a name for a normal, projective variety where every effective divisor is ample?
Is there a name for a normal, projective variety such that every effective divisor is ample? Examples of such varieties are projective space, weighted projective spaces, and simple Abelian varieties ...
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Example of a ruled, CM, $ \mathbb{Q} $-factorial, normal, Mori dream space whose Cox ring is integral but not CM,
This question is related to one I asked here in Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM. In ...
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Kodaira dimension of spaces of rational curves in hypersurfaces
Let $X\subset\mathbb{P}^n$ be a general hypersurface of degree $d\leq n$, and $\overline{\mathcal{M}}_{0,0}(X,a)$ the Kontsevich space of degree $a$ rational curves in $X$.
Does there exist an ...
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"Vanishing locus" of forms in the $h$-topology
Let $\Omega_{h}^p$ be the sheaf of $p$-forms in the $h$-topology defined as the sheafification for the $h$-topology of the presheaf,
$$ Y \mapsto \Omega^p_Y(Y) $$
Kebekus and Schnell show that when $X$...
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Question about algebraic curve being birational to smooth projective curve
Let $X$ be a geometrically irreducible affine variety defined over $\mathbb{Q}$ and dimension $1$. Then it is known that $X$ is birational over $\mathbb{C}$ to a smooth projective curve $C$.
I was ...
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55
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Composition of correspondences pulled back to $\mathrm{CH}_0$
Let $X,Y,Z$ be varieties. Given two correspondences $\Gamma_1 \subset X \times Y$ and $\Gamma_2 \subset Y \times Z$ there is a composition,
$$ [\Gamma_1] \circ [\Gamma_2] = \pi_{13 *} (\pi_{12}^* [\...
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Flips and Flops in minimal model program
Is there a concrete geometric intuition behind flips and flops in context of minimal model program one should keep in mind? Wikipedia says that these can be considered as algebraic analoga of ...
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Is the Cox ring of a $ \mathbb{Q} $-factorial, $ F $-regular, Mori dream space $ F $-regular?
A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...
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When is a smooth point of a projective, simplicial, toric variety $ X_{\Sigma} $ compatibly $ F $-split?
A variety $ X $ is $ F $-split if there exists an $ \mathcal{O}_{X} $-linear map $ \phi: F_{\ast}(\mathcal{O}_{X}) \to \mathcal{O}_{X} $ such that $ \phi \circ F^{\sharp} = \operatorname{id}_{\mathcal{...
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Does one only need to look at torus invariant curves to calculate the Seshadri constant for a point of a toric variety?
If $ X $ is an irreducible projective variety, $ L $ is a Nef divisor on $ X $, $ x $ is a point of $ X $, and $ \pi: \operatorname{Bl}_{x}(X) \to X $ is the natural projection morphism, then the ...
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Analogous tensor product operation for reflexive sheaf
Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it.
Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...
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$0$-dimensional intersection in weighted projective space
Consider homogeneous polynomials $P_0,P_1,P_2,P_3,P_4,P_5$ of degrees $3,3,2,3,2,1$ over $\mathbb{P}^3$, and the map $\phi:\mathbb{P}^3\rightarrow\mathbb{P} = \mathbb{P}(3,3,2,3,2,1)$ given by
$$
\phi(...
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Confusion with terminology: Crepant resolution of terminal singularities
In Theorem 1.1 of this article, Bridgeland proves derived equivalence between Crepant resolution of threefold terminal singularities. I am a little confused with this terminology. In particular, a $\...
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Does anyone know a rationally chain connected, Cohen Macaulay variety which is not separably rationally connected?
An $ n $-dimensional variety (here variety means an integral, separated, scheme of finite type over an algebraically closed field) $ X $ over a field $ k $ is rational if there is a birational map $ \...
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Rational points on surfaces
Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form
$$
S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\}
$$
where $...
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On the exceptional divisor of the resolution of indeterminacy locus of rational map
Let $f:X \dashrightarrow Y$ be a rational map between smooth, projective varieties over $\mathbb{C}$. We know that there is a resolution of the indeterminacy locus using which we obtain a smooth, ...
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Fibers of period map
Consider the period map for some smooth varieties, $p:\mathcal{X}\rightarrow\mathcal{A}:X\mapsto J(X)$, where $\mathcal{X}$ is moduli space of certain varieties and $\mathcal{A}$ is moduli space of ...
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descent of nef divisors
Let $f:X \rightarrow Y$ be a flat surjective morphism of complex projective varieties with connected fibers, $X$ normal, $Y$ smooth and $dim (Y) < dim(X)$. Suppose $L \in Pic (X)$ such that $L|_{X_{...
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Cone and contraction theorems for certain sub-klt pairs
Suppose $(Y,B_Y)$ is a sub-klt pair which is a birational model of a klt pair $(X,B)$: there exists a birational morphism $\pi:(Y,B_Y) \rightarrow (X,B)$ such that $\pi^{*}(K_X+B)=K_Y+B_Y$. We know ...
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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 ...
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Blow up along codimension 1 subscheme
Let $X$ be a smooth projective variety. $Z$ be a closed subscheme of codimension $1$ (potentially with embedded points). $Z_{red}$ is a Cartier divisor and $Bl_{Z_{red}}X$ is just $X$. What about the ...
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Non-uniruled connected smooth fibers implies flat
Let $f:X\to Y$ be a surjective morphism of connected smooth projective varieties over an algebraically closed field.
Assume all fibers are connected smooth and none are uniruled. Is $f$ flat?
In ...
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144
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An analogue of Noether's Problem for non-rational varieties
For the sake of simplicity, let $\mathsf{k}$ be algebraically closed and of zero characteristic. Varieties are irreducible.
The (linear) Noether's Problem (which goes back to the early 1910's in ...
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201
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Clemens-Griffiths component birational invariant
Let $X$ be a smooth variety over complex numbers $\mathbb{C}$, say a threefold for sake of better intuition. Is there any geometrical intuition behind the fact that
the Clemens-Griffiths component of ...
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144
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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 ...
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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 ...
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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 ...
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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 $X$ ...
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153
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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?
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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 ...
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Is there a way to explicitly find any rational $\mathbb{F}_p$-curve on the Kummer surface?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$)
$$
E\!:y_1^2 = ...
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Rational curves on ruled surfaces
Let $S$ be a ruled surface (over an algebraically closed field) with an $\mathbb{P}^1$-bundle $\pi\!: S \to E$ onto an elliptic curve $E$. What is the classification of (possibly singular) irreducible ...
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86
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Extending linear algebraic group action
Suppose we have a linear algebraic group $G$ acting birationally on a complete variety $X$ i.e. we are given a rational morphism $G\times X\dashrightarrow X$ satisfying obvious properties. Is there a (...
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140
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Cubic 3-fold singular along a curve
Does there exists a cubic or quartic $3$-fold $X\subset\mathbb{P}^4$ such that $Sing(X)$ is a smooth curve $C$ of genus $g(C)\geq 2$ and $X$ has $A_1$-singularities along $C$?
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Isometry fixes ample class implies it is an automorphism?
My question is related to lemma 5.12 in ["Normal Subgroups in the Cremona Group"]. Let $h$ be a birational transformation of a projective surface $X$ and $[D']\in N^1(X)$ be an ample class. We define $...
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Bounding the denominator in the canonical bundle formula
My question concerns with Theorem 3.1 in the paper "A canonical bundle formula" by Fujino and Mori.
The theorem claims the following:
Suppose $X \to C$ is a fiberation whose general fiber $F$ has ...
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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 ...
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Small contractions as blow ups
To improve my chances of getting answers/ comments I post my mathstackexchange question https://math.stackexchange.com/q/2808852/42548 also here.
I am trying to learn a bit about birational morphisms:...
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Liftability of varieties, after fpqc base change
Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable.
Suppose there exists an fpqc cover $S'\to S$, such ...
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153
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Euler form on three-fold
Let $X$ be a smooth projective $3$-fold over $\mathbb{C}$. Let $K_0(X)$ be its Grothendieck group, consider the Euler form defined as: $\chi(M,N): K_0(X)\times K_0(X)\rightarrow\mathbb{Z}$ by $(M,N)\...
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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_{...
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252
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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 $\...
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333
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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 ...
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How can I find a basis of $H^0(Y, -K_Y)$ for the del Pezzo surface $Y$ of degree 5?
Consider the surface $X \subset \mathbb{P}^2_{x, y, z}\times\mathbb{P}^1_{t, s}$:
$$
s^3y^2 + t^3yz = (t+s)x^2 + tsxz + t(t^2 +s^2)z^2
$$
over a finite field $k = \mathbb{F}_{2^d}$, $\mathrm{gcd}(d, 6)...
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266
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Desingularization of subvariety
Let $X$ be a smooth projective complex variety. Let $Y \subset X$ is a singular closed subvariety of $X$. Does there exist a birational morphism $\pi: \widetilde{X} \to X$, such that the proper ...
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145
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Purely inseparable $k$-rational dominant maps between an absolutely irreducible $k$-surface and $\mathbb{P}^2$
Let $X$ be an absolutely irreducible surface over an algebraically closed or finite field $k$ of characteristic $p$. Is it true that the following conditions are equivalent?
There is a purely ...
2
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108
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Birational map from even to odd degree curve. What is the image of one of infinity points?
Suppose I have a curve $C_1$ determined by $C_1: y^2 = (x+a)(f_{2g+1} x^{2g+1} + \dots + f_0)$. It has even degree polynomial of $x$ on the right side. I want to consider its image under birational ...
2
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269
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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 ...
2
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226
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How to prove two manifolds are not birational?
Given a family of compact complex manifold $\mathcal{X} \rightarrow B$, what are the standard techniques to prove two distinct fibers $\mathcal{X}_a$ and $\mathcal{X}_b$ are not birational?
2
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singularities preserved by integral closure
Let $X$ be an affine variety. Let $A$ be the coordinate ring of $X$ and let $K$ be the fraction field of $A$. Given a Galois extension $K\subset L$, let $B$ be the integral closure of $A$ in $L$. Let ...