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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Normality and integrality of schemes and splitting of map from structure sheaf to (derived)pushforward of structure sheaf along proper birational map

Let $R, S$ be commutative Noetherian rings such that $R$ is a subring of $S$. If $S$ is a normal domain, and there exists an $R$-linear map $\phi: S\to R$ whose restriction on $R$ is the identity map, ...
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Unirationality connected with $S$-unit equation

This update of question asked before. Let $n$ be a natural number. Consider a subvariety in $\mathbb A^{3n+2}$ (say over $\mathbb C$) given by the equation $$x_1(t-y_1)\dots (t-y_n)+x_2(t-z_1)\dots(t-...
Galois group's user avatar
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About the finite generation of log canonical rings in BCHM

I have posted this question on MSE but haven't received an answer yet. I rephrase it here. Let $(X,B)$ be a klt pair where $K_X+B$ is $\mathbb{R}$-Cartier. Let $\pi:X\rightarrow U$ be a projective ...
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About the semi-ampleness for $\mathbb{R}$-divisor

Let $X$ be a normal proper variety and $D$ an $\mathbb{R}$-Cartier divisor on $X$. Then $D$ is called a semi-ample $\mathbb{R}$-divisor if there is a morphism $f:X\rightarrow Y$ and a ample $\mathbb{R}...
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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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Nonequidimensional birational Mori contractions

I have been looking for an excplicit example of a birational, divisorial Mori contraction such that the exceptional locus is not equidimensional onto its image. To agree with the setup I like, the ...
p0lydactyl's user avatar
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Nice, concrete example of pl-flipping contraction

In a course I'm giving on the MMP, I am discussing the importance of Shokurov's notion of a pl-flipping contraction for showing that flips exist for arbitrary flipping contractions. Does someone have ...
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Picard group of a birational contraction

Let $X,Y$ be normal projective varieties and $f:X\to Y$ be a birational contraction, i.e. a birational morphism which satisfies $f_\ast O_X=O_Y$. Then $f^\ast:\text{Pic}(Y)\to \text{Pic}(X)$ is ...
Daebeom Choi's user avatar
1 vote
1 answer
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Very ample + effective = ample?

Sorry if this question is not appropriate for this site, but I haven't got an answer on stackexchange. It's well known that there are divisors (on a normal projective variety over the complex numbers) ...
Calculus101's user avatar
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Reference request showing that a very general Abelian variety $ A $ of genus $ g>1 $ has cyclic class group with ample generator

In Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM I asked for an example of a Cohen Macaulay, normal, ...
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Application of MMP in other branches of algebraic geometry

I'm learning minimal model program (MMP) recently. For a projective variety $X$, following MMP, we can do a sequence of birational transformations making $K_X$ nef or to a Mori fiber space. My ...
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Why the following quasi isomorphism implies the morphism to be a resolution (a step in the paper A characterization of rational singularities)

I was reading the paper A Characterization of Rational Singularities by Professor Kovács. The main theorem is stated as follows: THEOREM 1. Let $\phi: Y \rightarrow X$ be a morphism of varieties over ...
yi li's user avatar
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Unirational algebraic group scheme smooth

Let $G$ an unirational algebraic $k$-group over base field $k$ in sense of this book on Neron models, (ie separated $k$-group scheme of finite type). On page 310 is claimed that unirationality implies ...
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MMP for surfaces over a curve where the geometric generic fiber is a rational curve

I am looking for an explaination or an reference for the following fact: Let $\pi:X\rightarrow Z$ be a contraction from a smooth surface $X$ to a curve $Z$. Assume that the geometric generic fiber of ...
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Coefficients of members in a base-point free linear system

Let $\mathbb{K}\in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\}$ and let $D$ be a base-point free(or ample if it is necessary) $\mathbb{K}$-divisor on a normal projective variety. I have two questions: When $...
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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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A quick introduction to the birational classification of projective curves

To give you some personal background: I am a ring theorist, and most of my research focus on invariant theory of noncommutative rings. Recently I became interested in a certain problem that requires a ...
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Example of a $ \mathbb{Q} $-factorial, CM normal, projective, Mori dream space $ Z $ such that $ \operatorname{Cox}(Z) $ is integral and not CM

Does anyone know an example of a $ \mathbb{Q} $-factorial, normal, Cohen Macaulay, projective, Mori dream space $ Z $ over a field $ k $ of arbitrary characteristic such that the Cox ring of $ Z $ is ...
Schemer1's user avatar
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Literature for noncommutative birational invariants

Let $k$ be an algebraically closed field of zero characteristic. All fields under discussion are fields over $k$, and all division rings are division algebras over $k$. There is rich theory of ...
jg1896's user avatar
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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 ...
Sam Streeter's user avatar
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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 ...
Misha Verbitsky's user avatar
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1 answer
232 views

(Non-)Rationality of a certain quotient of the symmetric square of the Fermat sextic (quartic) curve

Consider the Fermat sextic curve $F: x^6 + y^6 + 1 = 0$ over an algebraically closed field of characteristic $0$. It has the two order $3$ automorphisms $\omega_x(x,y) := (\omega x, y)$ and $\omega_y(...
Dimitri Koshelev's user avatar
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135 views

Exceptional locus of proper birational morphism from smooth variety to normal variety

Let $f:Y\rightarrow X$ be a proper birational morphism. Suppose that $X$ is normal and $Y$ is smooth. Let us write the largest open subset U of X such that $f^{-1}$ can be defined. I want to show that ...
Irish's user avatar
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Proofs for if $\widetilde X\rightarrow X$ is a modification & $\widetilde X$ is a $\partial\bar{\partial}$-manifold, then so is $X$

A celebrated result due to Deligne--Griffiths--Morgan--Sullivan (see Theorem 5.22 in Real Homotopy Theory of Kaehler Manifolds) says that: Consider a proper modification $f:\widetilde M\rightarrow M$ ...
Invariance's user avatar
4 votes
1 answer
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Finitely generated section ring of Mori dream spaces

Set-up: We work over $\mathbb{C}$. Let $X$ be a Mori dream space. Define, following Hu-Keel, the Cox ring of $X$ as the multisection ring $$\text{Cox}(X)=\bigoplus_{(m_1\ldots,m_k)\in \mathbb{N}^k} \...
OrdinaryAttention's user avatar
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1 answer
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Explicit functor from Kuznetsov component to derived category of K3 for rational cubic fourfolds

Let $X \subset \mathbb{P}^5$ be a Pfaffian cubic fourfold (or one of the other known rational cubic fourfolds). It is known by Kuznetsov's Homological Projective Duality that $\mathcal{K}u(X) \simeq D^...
mathphys's user avatar
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2 answers
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Proper birational morphism from a Gorenstein normal scheme to a normal local domain, with trivial higher direct images, implies Cohen-Macaulay?

Let $k$ be a field of characteristic $0$. Let $R$ be a Noetherian local normal domain containing $k$. Also assume that $R$ is the homomorphic image of a Gorenstein ring of finite dimension, hence $R$ ...
Snake Eyes's user avatar
1 vote
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Dyck's Theorem via a Birational Transformation

In our paper, at Journal of Geometry and Graphics, v. 20, n. 1 (2016), Danny Maienschein and I showed that some well-known transformations of the real projective plane (defined almost everywhere), ...
Michael Rieck's user avatar
8 votes
1 answer
321 views

Alterations and smooth complete intersections

Let $k$ be an algebraically closed field, and $X$ a projective variety over $k$. Let $i : X\subset \mathbf{P}^d_k$ be a closed immersion into a projective space of high enough dimension. Is there a ...
user avatar
2 votes
1 answer
159 views

Cohen-Macaulay fiber products

Let $R$ be a regular local ring, $X$ and $Y$ smooth $R$-schemes, $T\to Y$ a regular closed immersion over $R$ with $T$ smooth over $R$, and $f: X\to Y$ an $R$-morphism. Is the fiber product scheme $...
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2 votes
1 answer
189 views

Inclusion of (pulling back of) dualizing sheaves under normalization

I am reading O. Fujino's book Iitaka Conjecture. In page 42, Lemma 3.1.19 he restated one result due to Viehweg to use the base change arguments. There exists some details in the proof of Step 2 in ...
Invariance's user avatar
6 votes
1 answer
348 views

Derived categories of smooth proper varieties?

We know several amazing techniques about the derived category $Perf (X)$ of a smooth projective variety such as the whole theory of Fourier-Mukai transforms. On the other hand, from a dg-categorical ...
P. Usada's user avatar
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0 answers
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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 ...
Puzzled's user avatar
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2 votes
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Extending étale covers from the regular locus to a resolution of singularities

Let $X$ be a normal proper variety with rational singularities (or terminal if that is necessary) and $X_{\text{reg}} \to X$ the regular locus. Let $\pi : \tilde{X} \to X$ be a resolution of ...
Ben C's user avatar
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3 votes
0 answers
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What are the possibilities of the general fibres in an Iitaka fibration?

This question is motivated by complex algebraic geometry. If $X$ is a complex algebraic variety with Kodaira dimension in $[1,\dim X-1]$, then the Iitaka fibration (the rational map induced by the ...
LeechLattice's user avatar
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2 votes
1 answer
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A Fourier-Mukai kernel locally given by a graph of a birational map and compatibility with extension

Let $X$ and $Y$ be smooth projective complex varieties. Suppose we have a Fourier-Mukai equivalence $$ \Phi_\mathcal P :Perf X \to Perf Y $$ with kernel $\mathcal P$. Moreover, suppose $\mathcal P$ ...
P. Usada's user avatar
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1 vote
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Modifying the base of a rational map

Let $f : X \dashrightarrow S$ be a rational map of smooth projective varieties. Is it true that, after a birational modification of $S$, every fiber intersects the domain of definition? Explicitly, is ...
Ben C's user avatar
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2 votes
0 answers
120 views

"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 $X$...
Ben C's user avatar
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4 votes
0 answers
204 views

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 ...
Ben C's user avatar
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2 votes
0 answers
156 views

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 ...
Johnny T.'s user avatar
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2 votes
1 answer
229 views

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 ...
user avatar
2 votes
2 answers
187 views

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 ...
Puzzled's user avatar
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3 votes
0 answers
108 views

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 ...
Ben C's user avatar
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2 votes
0 answers
55 views

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}^* [\...
Ben C's user avatar
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2 votes
1 answer
349 views

$K3$ 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 ...
user267839's user avatar
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1 vote
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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 ...
anonymous's user avatar
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0 answers
161 views

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 ...
user267839's user avatar
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2 votes
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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 ...
user267839's user avatar
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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 ...
Schemer1's user avatar
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4 votes
1 answer
232 views

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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