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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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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What is the obstruction to uniruledness being uninteresting?

An $ n $-dimensional projective variety $ Z $ over a field $ k $ (assume characteristic zero) is uniruled if there is an $ n-1 $-dimensional variety $ Y $ and a dominant, generically finite, rational ...
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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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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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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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2 answers
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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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Genus of algebraic curves

Let $X$ be an integral (possibly singular) projective algebraic curve of degree $d$ in $\mathbf{P}^n_{K}$, where $K$ is a field that is either the real numbers, the complex numbers, or a field that is ...
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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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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
357 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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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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4 votes
3 answers
217 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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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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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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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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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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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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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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2 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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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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Question regarding linear system of projective space

I am currently reading the paper titled "Birational Geometry of Moduli spaces of Configurations of Points on the Line" by M.Bolognesi and A.Massarenti. I have following doubts in section 2....
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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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4 votes
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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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1 answer
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Sheaves on families of genus 2 curves in Hassett's paper

Sorry for a maybe stupid long question but I'm reading the paper "Classical and minimal models of the moduli space of curves of genus two" by Brendan Hassett and I'm not able to unravel a ...
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5 votes
1 answer
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Equivalent definitions of Kodaira dimension

The Kodaira($-$Iitaka) dimension of a line bundle $L$ on a complex manifold $X$ can be defined either in three ways: The maximal dimension of the image of the rational maps $φ_{|mL|} : X \...
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Positivity of the global log canonical threshold of a pair

Let $(X,L)$ be a polarized smooth projective variety. Let $D$ be a smooth irreducible divisor in $X$. Let $0<c<1$ be a real number. We denote $cD$ as $\Delta$. We can define the $\alpha$ ...
7 votes
1 answer
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Singular curves of genus 1

Let $C$ be an irreducible curve of arithmetic genus $1$ over a field $k$ and with a double $k$-point $p\in C$. Is $C$ rational over $k$? If $C$ is a plane cubic the answer is positive since we can ...
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Volume of conic bundles

Consider a smooth conic bundle $X\rightarrow \mathbb{P}^1$ with discriminant of degree $d$ (the locus of $\mathbb{P}^1$ over which the fibers are reducible conics). There is a formula for $(-K_X)^2$ ...
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1 answer
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Confusion about the (Grothendieck–Poincaré) double dual of reflexive differentials vs usual differentials on a normal Cohen–Macaulay scheme

$\DeclareMathOperator\Hom{Hom}$Let $\mathcal{A}$ be an abelian category, my question is about the case when $\mathcal{A}$ is the category of quasi-coherent sheaves on a scheme $X$. There is a fully ...
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4 votes
2 answers
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Smoothness of fibers over finite fields

Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties over a finite field of characteristic different from $2$. Is there any result on the existence of a point $y\in Y$ such that $X_y = ...
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2 votes
0 answers
136 views

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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4 votes
1 answer
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Del Pezzo surfaces of degree four and complete intersections of two quadrics

Let $X = Q_1\cap Q_2$ be a complete intersection of two smooth quadrics, over a field $K$, in $\mathbb{P}^4$ with homogeneous coordinates $y_0,y_1,y_2,y_3,y_4$. Set $Q_1 = \{F_1 = 0\}$ and $Q_2 = \{...
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2 answers
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Smooth complete intersections

Let $X_{2,3}\subset\mathbb{P}^n$, with $n\geq 5$, be a complete intersection of a quadric $X_2$ and a cubic $X_3$ containing a $2$-plane $H$. Assume $X_2$ and $X_3$ to be general among the ...
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1 vote
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Why do such a birational map exists? And why it is unique?

Let $G$ be a complex linear algebraic group which is connected and reductive and let $\mathfrak g$ be its Lie algebra. Suppose that $H \subset G$ is a 1-dimensional torus such that the action of $H$ ...
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