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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Flat scheme-theoretic closure
Suppose $R$ is a discrete valuation ring with fraction field $K$. Let $X\subset \mathbf{P}^n_{C_K}$ be a closed subscheme, flat over $C_K$, a smooth projective curve over $K$.
Let $C_R$ be a flat ...
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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 ...
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Torsion of Fermat hypersurfaces
An interesting invariant of a rationally chain-connected variety $X/k$ is the exponent of the group,
$$ \ker{(\mathrm{CH}_0(X_K) \xrightarrow{\deg} \mathbb{Z})} $$
where $K = k(X)$ is the function ...
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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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$K3$ surfaces can't be uniruled
Let $S$ be a uniruled surface, ie admits a dominant map $ f:X \times \mathbb{P}^1$. Why then it's canonical divisor $\omega_X$ cannot be trivial? Motivation: I want to understand why $K3$ surfaces ...
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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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Singularities of toric pairs
Suppose $(X,B)$ is a log canonical pair and $f: X \rightarrow Y$ an equidimensional toroidal contraction such that every component of $B$ is $f$ -horizontal. Let $\Gamma$ denote the reduced ...
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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 there some kind of construction of a "canonical unirational variety" like the one for toric varieties?
Toric varieties in some sense a "canonical rational variety" in that one can construct them from purely combinatorial data and this combinatorial data makes it possible to turn many ...
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Cycles contained in ample enough hypersurfaces
Let $X$ be an irreducible smooth projective variety of pure dimension $d$ over the complex numbers and $Z\subset X\times X$ a codimension $d$ irreducible smooth closed subvariety.
Is there a smooth ...
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If $ Z $ is an $ n $-dimensional, projective variety, containing $ \mathbb{G}_{m}^{n} $, what is the obstruction to $ Z $ being toric?
Let $ Z $ be an $ n $-dimensional, projective, variety, over a field of arbitrary characteristic and let $ \iota: \mathbb G_{m}^{n} \to Z $ be a morphism such that for any $ z \in Z $, the fibre $ \...
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Why $\overline{\operatorname{Mov}(X)}\subset \overline{\operatorname{Eff}(X)}$, where $X$ is any normal variety?
I am trying to understand why $\overline{\operatorname{Mov}(X)}\subset \overline{\operatorname{Eff}(X)}$, where $X$ is any normal variety.
Here $\operatorname{Mov}(X)$ is the convex cone generated by ...
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A variant on the Fujita invariant
Let $X$ be a Fano variety over $\mathbb{C}$. Let $D$ be a divisor on $X$. Recall that the Fujita invariant of $D$ is defined to be
$$a(D) = \inf \{ t \in \mathbb{R} : K_X + tD \text{ is effective} \}.$...
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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 ...
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Blow-ups of $ F $-regular varieties at points in general position and finite generation of the Cox ring
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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Example of projective, $ F $-regular variety $ X $ and smooth sub-variety $ Y $ such that $ \operatorname{Bl}_{Y}(X) $ is not $ 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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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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Normal bundle of a linear subspace
Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ ...
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Does nefness in analytic setting depend on Hermitian metric?
I'm reading Demailly's book 'Analytic Methods in Algebraic Geometry'.
Let $X$ be a compact complex manifold with a Hermitian metric. A line bundle $L$ is said to be nef if for every $\epsilon>0$, ...
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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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Determine the coefficient of the exceptional divisor
Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X_0$ be a reduced (Cartier) ...
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Do we have a simple proof for this criterion for basepoint freeness?
The following is a criterion by Fujita (On the structure of polarized varieties with $\Delta$-genera zero).
Consider a complex smooth projective variety $X$ of dimension $n$ and an ample divisor $H$, ...
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Classification of quartic surfaces
Let $k$ be a field of characteristic zero (non necessarily algebraically closed, we may assume for instance that $k = \mathbb{C}(t)$). Does there exist a classification of degree four surfaces $S\...
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Current progress on rationality problem for complex hypersurfaces
How is the current progress on rationality problem for complex hypersurfaces $X\subset\mathbb{P}^{n+1}$ with $n\geq 3$?
There are many hypersurfaces are shown to be unrational, such as smooth cubic ...
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Geometry of contracted divisors
Let $f:\mathbb{P}^3\dashrightarrow\mathbb{P}^2$ be a dominant rational map defined over a field $k$ (not necessarily algebraically closed) of characteristic zero.
Consider a resolution $\widetilde{f}:...
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How to construct log-canonical (or Calabi-Yau), non-Cohen-Macaulay singularities of low codimensions?
(EDIT 07/06/11: although the question has not been settled definitely, Sándor's excellent answer and the comments by Angelo and ulrich have highlighted many potential obstructions to the constructions ...
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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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Topology of the preimage of a point for degree one holomorphic maps
Let $M^n$ and $N^n$ be two compact complex (or complex projective) manifolds. Let $f: M\to N$
be a holomorphic map of degree one. How to prove that for each $x\in N$ the set
$f^{-1}(x)$ is simply-...
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For what properties $ (\mathcal{P}) $ (if any) does $ (\mathcal{P}) $ + analytic isomorphism imply birationality?
In general if $ \mathfrak{p} \in \operatorname{Spec}(A) $ and $ \mathfrak{q} \in \operatorname{Spec}(B) $ are two (not necessarily closed) points such that $ (\mathfrak{p}, \operatorname{Spec}(A)) $ ...
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What is the minimal $ n $ such that the Cox ring of the blow up of a simplicial, $ r $-dimensional, toric variety at $ n $ points in g.p. is not f.g.?
In Shigeru Mukai's paper "Counterexample to Hilbert's 14th Problem for the 3-dimensional Additive Group," Mukai proved that if $ \frac{1}{r+1}+\frac{1}{n-r-1} \le \frac{1}{2} $, then the ...
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Does analytic isomorphism imply local isomorphism?
If $ \mathfrak{p} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(A) $, and $ \mathfrak{q} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(B) $ such ...
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How to link the rank of Elliptic Surfaces and Elliptic Curves over function fields?
I have been investigating certain elliptic surfaces for my research, and when giving a presentation, I was asked why when given an elliptic surface $E$, say given in Weierstrass form
$$E\colon y^2+a_1(...
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Possible singularities of the base of a Mori fiber space
Suppose X is a normal projective complex variety, (X, $\Delta$) is a klt pair and f : X $\to$ Z is a Mori fiber space given by a contraction of an extremal ray for this pair. Here I mean that the ...
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Every locally free sheaf is Cohen-Macaulay on complex variety with at canonical singualities?
Suppose $X$ is a normal complex space with at most singularities, can we say any locally free sheaf on it is Cohen-Macauly?
Recall that a coherent sheaf $\mathcal{F}$ over $X$ is called (maximal) ...
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Relative minimality for conic bundles
Due to difficulties in finding references I am not sure of what relative minimality is about for conic bundles. Suppose we're working on con. bun. over surfaces.
The definition is ok: the fiber over ...
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In complex analytic category, is the pluricanonical sheaf Cohen--Macaulay?
We adopt the following definition of canonical singularities in complex analytic category.
Let $X$ be a normal complex space of dimension $n$, and let $j:X_{\text{reg}}\rightarrow X$ be the open ...
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Two morphisms possess the same Viehweg's variation
Recall the definition of Viehweg's variation intimated from Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fibre Spaces,
E. Viehweg
Let $f: V\rightarrow W$ be a fiber space (...
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Rational curves on the image of the pluricanonical maps
Let $X$ be a compact complex manifold with canonical bundle $K_X$. Assume the Kodaira dimension $\kappa(X)$ is positive (but not maximal, i.e., $X$ is not of general type). Let $\varphi_m : X \...
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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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Intersection pairing and birational morphisms
Let $f:X\to Y$ be a birational morphism of smooth projective variety. We assume that $f(V)\simeq U$ isomorphism induced by $f$, where $V\subset X$ and $U\subset Y$ are two Zariski open sets. Let $x\in ...
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Blowing up of a singular subvariety
I ask the same question on MathStackExchange but receive no answer.
I'm reading Kollár Mori Chapter 2.3. And they state the following lemma:
Lemma 2.29. Let $X$ be a smooth variety and $\Delta = \sum ...
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Numerical reduction map for line bundles?
For a nef line bundle $L$ on a normal projective variety $X$, we have three invariants- the nef dimension $n(L)$, the numerical dimension $\nu(L)$ and the Iitaka dimension $\kappa(L)$. $n(L)$ is ...
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Do blowups generate the birational equivalence relation?
Suppose $X$ and $Y$ are birational varieties (projective, irreducible, over $\mathbb C$) of the same dimention. Is there a sequence of blow-ups and blow-downs that makes $Y$ from $X$?
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How to understand flip intuitively?
In MMP, when we get a small extremal contraction $f:X\rightarrow Y$, we will flip it to $f^+:X^+\rightarrow Y$ such that $K_{X^+}+D^+$ is $f^+$-ample. Technically, I understand that is because we want ...
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Discrepancy of a divisor over a different model
I also asked this question on MathStackExchange but receive no answers.
I'm reading Koll'ar and Mori's book about singularity theory. They state the following lemma without proof:
Lemma 2.30. Let $f:...
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Isomorphism outside of negative curves against the canonical
Let $X$ be a smooth projective complex variety and let us suppose that the closure of the union of curves $C$ on $X$ that are non-positive against the canonical divisor is a closed subset $F\subsetneq ...
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Blowups of Cohen-Macaulay varieties
Suppose that $X$ is a Cohen-Macaulay normal scheme/variety and $\pi : Y \to X$ is a proper birational map with $Y$ normal.
Question: Is $Y$ also Cohen-Macaulay? Are there common conditions which ...
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Is normality really needed for the Cox ring of a $ \mathbb{Q} $-factorial variety to be well defined or is regular in codimension one enough?
I recently was looking at Chapter 1, Section 4.1 of the book on Cox Rings by Ivan Arzhantsev, Ulrich Derenthal, Jurgen Hausen, and Antonio Laface (see https://arxiv.org/pdf/1003.4229.pdf) I noticed ...
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When does $\operatorname{Aut}(X)=\operatorname{Bir}(X)$ hold?
Let $X$ be a projective complex manifold. Under what condition do we have the equality $\operatorname{Aut}(X)=\operatorname{Bir}(X)$? Here $\operatorname{Aut}(X)$ denotes the group of holomorphic ...
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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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