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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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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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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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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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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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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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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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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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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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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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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1 vote
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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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1 vote
1 answer
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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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2 votes
1 answer
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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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1 vote
1 answer
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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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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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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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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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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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6 votes
1 answer
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Iitaka dimension is invariant under surjective morphism between smooth projective varieties

I would like to prove the following result (working on $\mathbb{C}$) but get trouble with the other direction. Let $f:Y'\rightarrow Y$ be a surjective morphism between smooth projective varieties, ...
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5 votes
1 answer
421 views

Example illustrating necessity of considering birational equivalence and not biholomorphic equivalence in MMP

The minimal model program attempts to classify algebraic varieties up to birational equivalence. For compact Riemann surfaces, Riemann's uniformization theorem tells us that the geometry of the curve ...
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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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3 votes
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Noether–Enriques using Tsen's lemma

Consider the following weak version of the Noether–Enriques theorem (field is $\mathbb{C}$): Let $\varphi:X\rightarrow Z$ be a morphism from a smooth projective surface onto a smooth curve with $F_z:=...
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3 votes
3 answers
263 views

Varieties with few trisecant lines

Let $X\subset\mathbb{P}^N$ be an irreducible projective variety. Let's denote by $\mathcal{T}$ the following property: through a general point $x\in X$ there is no line intersecting $X$ in at least ...
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1 vote
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Birational morphisms from DM stacks to their coarse moduli spaces

Let $X$ be an integral scheme over a field. Let $G$ be a finite group acting on $X$ faithfully. Assume the quotient stack $[X/G]$ is separated (e.g., when $G$ acts on $X$ properly). Then $[X/G]$ is a ...
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3 votes
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Relation between closed cone of curves of a small $\mathbb{Q}$-factorial modification of Mori dream space

Let $X$ be a smooth projective variety which is a Mori dream space. Then we know there are finite small $\mathbb{Q}$-factorial modifications $f_i:X\dashrightarrow X_i$ for $i=1,\cdots,k$ such that $...
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4 votes
0 answers
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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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3 votes
0 answers
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Blowing-up a non reduced fiber

Let $X\rightarrow \mathbb{P}^2$ be a smooth conic bundle with a non reduced fiber $F$, and $\widetilde{X}$ the blow-up of $X$ along $F$ with exceptional divisor $F\times\mathbb{P}^1$. I expect $\...
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7 votes
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Field extensions over which algebraic varieties cannot acquire points

The following fact (slightly reworded here) is proven in this answer: If $K$ is a purely transcendental extension of an infinite¹ field $k$, then whenever a separated scheme $X$ of finite type over $...
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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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1 vote
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Surfaces with rational double points

Let $S\rightarrow \mathbb{P}^1$ a surface fibered in conics over a field. Assume that $S$ has a single non reduced fiber $F$ with two points of type $A_1$ on it. Blowing-up the two points and ...
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3 votes
1 answer
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Singularities of surfaces fibered in rational curves

Let $S$ be a projective surface with a morphism $S\rightarrow\mathbb{P}^1$ whose fibers are either smooth $\mathbb{P}^1$'s or the union of two smooth $\mathbb{P}^1$'s intersecting in a point. ...
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3 votes
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Description of movable cone

Let $X$ be a normal, $\mathbb{Q}$-factorial projective variety (over $\mathbb{C}$). If we assume that $X$ is a Mori dream space, then by definition its movable cone is rational polyhedral, and there ...
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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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7 votes
1 answer
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Is there a classification of minimal algebraic threefolds?

The minimal model program aims to find a minimal representative in the birational class of a given variety with reasonable singularities. Assuming this has been done, it seems natural to ask what ...
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Birationally equivalent elliptic curves and singularities

I got the following cubic elliptic curve from some physical problem $$E_c(\mathbb{C}): w^2=4 z^3-zG_2-G_3,$$ where $G_2=3 \alpha ^2+\gamma$ and $G_3=\alpha ^3-\alpha \gamma -\beta ^2$ for known ...
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