Derived Categories and Birational Geometry

Derived categories and birationality

We discuss the question of finding conditions on a derived equivalence between two smooth projective varieties $X$ and $Y$ that imply that $X$ and $Y$ are birational. The types of conditions we consider are in the spirit of finding categorical analogous of classical Torelli theorems. We study, in particular, a notion of strongly filtered derived equivalence and study cases where strongly filtered derived equivalence implies birationality. We also consider an open variant of our main question.

Flops and derived categories

This paper contains some applications of Fourier-Mukai techniques to the birational geometry of threefolds. In particular, we prove that birational Calabi-Yau threefolds have equivalent derived categories. To do this we show how flops arise naturally as moduli spaces of perverse coherent sheaves.

等变理性几何中的不变量

Invariants in equivariant birational geometry

本文研究了等变理性几何中不变量的性质与应用。

Torsors and stable equivariant birational geometry

We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups.

Deligne-Mumford 栈的双有理几何

Birational geometry of Deligne-Mumford stacks

本文研究 Deligne-Mumford 栈的双有理几何，并定义了新的双有理不变量。

Equivariant birational geometry of linear actions

We study linear actions of finite groups in small dimensions, up to equivariant birationality.

Symbols and equivariant birational geometry in small dimensions

We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions.

Exercises in the birational geometry of algebraic varieties

This a collection of about 100 exercises. It could be used as a supplement to the book Kollár--Mori: Birational geometry of algebraic varieties.

The Lefschetz principle in birational geometry: birational twin varieties

Inspired by the Weak Lefschetz Principle, we study when a smooth projective variety fully determines the birational geometry of some of its subvarieties. In particular, we consider the natural embedding of the space of complete quadrics into the space of complete collineations and we observe that their birational geometry, from the point of view of Mori theory, fully determines each other. When two varieties are related in this way, we call them birational twins. We explore this notion and its various flavors for other embeddings between Mori dream spaces.

准线性作用于del Pezzo表面的理查德几何

Birational geometry of actions on del Pezzo surfaces

本文完成有限群在del Pezzo表面的正则一般自由作用的分类，解决了若干等变理查德几何中的开放问题。

Birational geometry of Fano double covers

We prove divisorial canonicity of Fano double hypersurfaces of general position.

Birational geometry of algebraic varieties

This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.

Birational geometry of defective varieties

Here we investigate the birational geometry of projective varieties of arbitrary dimension having defective higher secant varieties. We apply the classical tool of tangential projections and we determine natural conditions for uniruledness, rational connectivity, and rationality.

Birational geometry

This article contains the notes of a graduate course on birational geometry focusing on the minimal model program. Topics covered include singularities, vanishing, nonvanishing, cone and contraction, base point freeness, finite generation, flips, termination, minimal models and Mori fibre spaces.

Witt环、Pfister型与等变双有理几何

Witt rings, Pfister forms, and equivariant birational geometry

该研究针对带有限群作用的二次型的等变双有理几何，发展了Witt群与Pfister理论的等变类似物，拓展了双有理几何的研究范畴。

Birational geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification

We determine the birational geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification, by looking at the locus of non-locally-free sheaves on the relevant moduli space.

Birational geometry of singular Fano varieties

We prove divisorial canonicity of Fano hypersurfaces and double spaces of general position with elementary singularities.

Birational geometry of the moduli space of pure sheaves on quadric surface

We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial $5m + 1$ and $c_{1} = (2, 3)$. We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.

Motivic measures and stable birational geometry

We study the motivic Grothendieck group of algebraic varieties from the point of view of stable birational geometry. In particular, we obtain a counter-example to a conjecture of M. Kapranov on the rationality of motivic zeta-function.

Birational geometry of irreducible holomorphic symplectic tenfolds of O'Grady type

In this paper we analyse the birational geometry of O'Grady ten dimensional manifolds, giving a characterisation of Kaehler classes and lagrangian fibrations. Moreover, we study symplectic compactifications of intermediate jacobian fibrations of smooth cubic fourfolds.

等变双有理几何中的线性化概念

Linearizability notions in equivariant birational geometry

本文讨论了有限群作用在代数簇上的双有理性质、线性化、torsors和versality。

Birational geometry of Beauville-Mukai systems II: general theory in low ranks

Via wall-crossing, we study the birational geometry of Beauville-Mukai systems on K3 surfaces with Picard rank one. We show that there is a class of walls which are always present in the movable cones of Beauville-Mukai systems. We give a complete description of the birational geometry of rank two Beauville-Mukai systems when the genus of the surface is small.

Birational Geometry of Rationally Connected Manifolds via Quasi-lines

This is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in ${\mathbb P}^n$ ({\it quasi-lines}). Various characterizations of a Zariski neighbourhood of a line in ${\mathbb P}^n$ are obtained, some of them being new. Also, methods of formal geometry are applied for deducing results of birational nature.

The birational geometry of moduli space of sheaves on the projective plane

We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves $M_H(v)$ on $\bb{P}^2$ and mini-wall crossings in the stability manifold $Stab(D^b(\bb{P}^2))$.

Generalised pairs in birational geometry

In this note we introduce generalised pairs from the perspective of the evolution of the notion of space in birational algebraic geometry. We describe some applications of generalised pairs in recent years and then mention a few open problems.

稳定化的关联度量：比较不同维数簇的双有理几何

Stabilized measures of association: comparing the birational geometry of varieties having different dimensions

本文推广了Lazarsfeld-Martin的关联度量，得到比较不同维数簇双有理几何的不变量，并给出实例与计算。

Equivariant birational geometry of quintic del Pezzo surface

In this paper we prove that there are exactly two $G$-minimal surfaces which are $G$-birational to the quintic del Pezzo surface, where $G \cong C_5 \rtimes C_4$. These surfaces are the quintic del Pezzo surface itself and the surface $\mathbb{P}^1 \times \mathbb{P}^1$.

Birational geometry of Fano double spaces of index two

We study birational geometry of Fano varieties, realized as double covers $σ\colon V\to {\mathbb P}^M$, $M\geq 5$, branched over generic hypersurfaces $W=W_{2(M-1)}$ of degree $2(M-1)$. We prove that the only structures of a rationally connected fiber space on $V$ are the pencils-subsystems of the free linear system $|-\frac12 K_V|$. The groups of birational and biregular self-maps of the variety $V$ coincide.

二面体在理性几何中的应用

Icosahedron in birational geometry

本文探讨了二面体群作用下射影和仿射空间的商结构，并关注其理性性质。

Frobenius techniques in birational geometry

This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry about the Frobenius type techniques recently used extensively in positive characteristic algebraic geometry. We first explain the basic ideas through simple versions of the fundamental definitions and statements, and then we survey most of the recent algebraic geometry results obtained using these techniques.

Birational geometry of terminal quartic 3-folds II: the importance of beings Q-factorial

The paper explores the birational geometry of terminal quartic 3-folds. In doing this I develop a new approach to study maximal singularities with positive dimensional centers. This allows to determine the pliability of a Q-factorial quartic with ordinary double points, and it shows the importance of Q-factoriality in the context of birational geometry of uniruled 3-folds

Birational geometry of the moduli space of rank 2 parabolic vector bundles on a rational curve

We investigate the birational geometry (in the sense of Mori's program) of the moduli space of rank 2 semistable parabolic vector bundles on a rational curve. We compute the effective cone of the moduli space and show that all birational models obtained by Mori's program are also moduli spaces of parabolic vector bundles with certain parabolic weights.

On the birational geometry for irreducible symplectic 4-folds related to the Fano schemes of lines

There is two group actions on the Fano scheme of lines such that the quotient becomes an irreducible symplectic manifold. We showed that both quotients are birational to the generalized Kummer variety or the 2-points Hilbert scheme of a K3 surface.

Gabriel's theorem and birational geometry

Extending work of Meinhardt and Partsch, we prove that two varieties are isomorphic in codimension c if and only if certain quotients of their categories of coherent sheaves are equivalent. This result interpolates between Gabriel's reconstruction theorem and the fact that two varieties are birational if and only if they have the same function field.

Derived categories view on rationality problems

We discuss a relation between the structure of derived categories of smooth projective varieties and their birational properties. We suggest a possible definition of a birational invariant, the derived category analogue of the intermediate Jacobian, and discuss its possible applications to the geometry of prime Fano threefolds and cubic fourfolds.

Birational geometry for d-critical loci and wall-crossing in Calabi-Yau 3-folds

The notion of d-critical loci was introduced by Joyce in order to give classical shadows of $(-1)$-shifted symplectic derived schemes. In this paper, we discuss birational geometry for d-critical loci, by introducing notions such as `d-critical flips', `d-critical flops', etc. They are not birational maps of the underlying spaces, but rather should be understood as virtual birational maps. We show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry. Among them, we show that wall-crossing diagrams…

Fano hypersurfaces and their birational geometry

We survey some results on the nonrationality and birational rigidity of certain hypersurfaces of Fano type. The focus is on hypersurfaces of Fano index one, but hypersurfaces of higher index are also discussed.

稳定退化与birational几何

Stable degeneration and birational geometry

本文探讨了代数K稳定性理论中稳定退化的最新进展，聚焦于birational几何中的关键问题与方法。

Birational geometry and the canonical ring of a family of determinantal 3-folds

Few explicit families of 3-folds are known for which the computation of the canonical ring is accessible and the birational geometry non-trivial. In this note we investigate a family of determinantal 3-folds in $\mathbb P^2 \times \mathbb P^3$ where this is the case.

Lectures on birational geometry

Lecture notes of a course on birational geometry (taught at College de France, Winter 2011, with the support of Fondation Sciences Mathématiques de Paris). Topics covered: introduction into the subject, contractions and extremal rays, pairs and singularities, Kodaira dimension, minimal model program, cone and contraction, vanishing, base point freeness, flips and local finite generation, pl flips and extension theorems, existence of minimal models and Mori fibre spaces, global finite generation, etc.

Birational geometry of terminal quartic 3-folds. I

This is the unabridged web version of the paper that will be published on the American Journal of Mathematics. In this paper, we study the birational geometry of certain examples of mildly singular quartic 3-folds. A quartic 3-fold is an example of a Fano variety, that is, a variety $X$ with ample anticanonical sheaf $Ø_X(-K_X)$. From the point of view of birational geometry they basically fall within two classes: either $X$ is ``close to being rational'', and then it has very many biregularly distinct birational models as a Fano 3-fold, or, at the other extreme, $X$ has a unique model and it …

Berkovich skeleta and birational geometry

We give a survey of joint work with Mircea Mustaţă and Chenyang Xu on the connections between the geometry of Berkovich spaces over the field of Laurent series and the birational geometry of one-parameter degenerations of smooth projective varieties. The central objects in our theory are the weight function and the essential skeleton of the degeneration. We tried to keep the text self-contained, so that it can serve as an introduction to Berkovich geometry for birational geometers.

GIT 商的双有理几何

The birational geometry of GIT quotients

针对 GIT 商双有理几何未被变异完全捕捉的问题，构造参数化所有双有理模型全部 GIT 商的空间，完整捕捉其双有理几何并给出稳定轨道集合的紧化。

Birational Geometry of 3-fold Mori Fibre Spaces

We study the geography and birational geometry of 3-fold conic bundles over P^2 and cubic del Pezzo fibrations over P^1. We discuss many explicit examples and raise several open questions. This paper was submitted to the proceedings of the "Fano conference" held in Torino in October 2002.

Computational birational geometry of minimal rational surfaces

The classification of minimal rational surfaces and the birational links between them by Iskovskikh, Manin and others is a well-known subject in the theory of algebraic surfaces. We explain algorithms that realise links of type II between minimal del Pezzo surfaces, one of the major classes of birational links, and we describe briefly how this fits into a large project to implement the results of Iskovskikh's programme in Magma.

On the birational geometry of varieties of maximal Albanese dimension

We study the birational geometry of varieties of maximal Albanese dimension. In particular we discuss criteria for a generically finite morphism of varieties of maximal Albanese dimension to be birational; we give a new characterization of Theta divisors; we study the Albanese map and refine some of the results of Kollár; finally we use these results to birationally classify varieties with $P_3(X)=2$ and $q(X)=dim (X)$. Our method combines the generic vanishing theorems of Green and Lazarsfeld, the theory of Fourier Mukai transforms and the results of Kollàr on higher direct images of dualizin…

旗簇的商及其双有理几何

Quotients of flag varieties and their birational geometry

研究四维复向量空间子空间完全旗簇的周商，通过计算证明其光滑且是森梦空间，详细描述双有理几何。

Why should a birational geometer care about Bridgeland stability conditions?

In this survey we borrow from Coskun and Huizenga an example of application of Bridgeland stability conditions to birational geometry and we rephrase it without assuming any previous knowledge about derived categories.

加权射影空间乘积中超曲面的双有理几何

Birational geometry of hypersurfaces in products of weighted projective spaces

本研究扩展了加权射影空间乘积中超曲面的双有理几何理论，确定了Mori梦想空间情形下的所有相关锥、双有理模型及Cox环，并证明了终端Calabi-Yau超曲面的Kawamata-Morrison锥猜想之双有理形式。

Stability conditions and birational geometry of projective surfaces

We show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.

层稳定对、Quot概形与双有理几何

Sheaf stable pairs, Quot-schemes, and birational geometry

本文建立层稳定对模空间理论与双有理几何的联系，重点研究光滑射影曲线上的稳定对模空间并给出低次数计算，同时提出若干后续问题。

On birational geometry of minimal threefolds with numerically trivial canonical divisors

For a minimal $3$-fold $X$ with $K_X\equiv 0$ and a nef and big Weil divisor $L$ on $X$, we investigate the birational geometry inspired by $L$. We prove that $|mL|$ and $|K_X+mL|$ give birational maps for all $m\geq 17$. The result remains true under weaker assumption that $L$ is big and has no stable base components.

Birational geometry of the Mukai system of rank two and genus two

Using the techniques of Bayer--Macrì, we determine the walls in the movable cone of the Mukai system of rank two for a general K3 surface $S$ of genus two. We study the (essentially unique) birational map to $S^{[5]}$ and decompose it into a sequence of flops. We give an interpretation of the exceptional loci in terms of Brill--Noether loci.

Birational geometry in codimension 2 of symplectic resolutions

We prove the conjecture that two projective symplectic resolutions for a symplectic variety $W$ are related by Mukai's elementary transformations over $W$ in codimension 2 in the following cases: (i). nilpotent orbit closures in a classical simple complex Lie algebra; (ii). some quotient symplectic varieties.