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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2 votes
1 answer
94 views

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, ...
2 votes
0 answers
89 views

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-...
1 vote
0 answers
81 views

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 ...
1 vote
1 answer
99 views

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}...
2 votes
0 answers
129 views

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 ...
1 vote
1 answer
104 views

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 ...
6 votes
1 answer
532 views

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}...
5 votes
1 answer
778 views

Definition of discrepancy

In Kollar and Mori "Birational geometry of algebraic varieties" discrepancy is defined as following way. Let X be a normal variety and $D = \sum_i a_i D_i$ be a $\mathbb{Q}$ divisor. Assume ...
1 vote
0 answers
104 views

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 ...
1 vote
0 answers
228 views

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 ...
1 vote
0 answers
153 views

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, ...
1 vote
1 answer
250 views

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) ...
7 votes
1 answer
474 views

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 ...
1 vote
0 answers
238 views

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 ...
21 votes
2 answers
2k views

Applications of derived categories to "Traditional Algebraic Geometry"

I would like to know how derived categories (in particular, derived categories of coherent sheaves) can give results about "Traditional Algebraic Geometry". I am mostly interested in classical ...
1 vote
1 answer
217 views

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 ...
2 votes
1 answer
121 views

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 ...
1 vote
0 answers
105 views

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 ...
1 vote
0 answers
93 views

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 $...
2 votes
0 answers
70 views

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 ...
1 vote
0 answers
144 views

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 ...
1 vote
0 answers
97 views

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 ...
1 vote
0 answers
95 views

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 ...
5 votes
0 answers
165 views

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 ...
4 votes
0 answers
85 views

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 ...
2 votes
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(...
0 votes
0 answers
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 ...
1 vote
0 answers
159 views

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 $f:\widetilde M\rightarrow M$ ...
4 votes
1 answer
136 views

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} \...
2 votes
1 answer
160 views

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^...
2 votes
2 answers
338 views

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 $R$ ...
1 vote
0 answers
44 views

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), ...
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 ...
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 $...
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 ...
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 ...
3 votes
0 answers
93 views

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 ...
2 votes
0 answers
131 views

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 ...
2 votes
0 answers
102 views

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 ...
2 votes
1 answer
128 views

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 $\mathcal P$ ...
1 vote
0 answers
113 views

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 ...
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 ...
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 $X$...
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 ...
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 ...
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 ...
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 ...
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}^* [\...
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 ...
5 votes
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 ...

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