Exercises in the birational geometry of algebraic varieties

On the birational automorphisms of varieties of general type

We show that the number of birational automorphisms of a variety of general type X is bounded by c \cdot \vol(X,K_X), where c is a constant which only depends on the dimension of X.

Finite $p$-groups of birational automorphisms and characterizations of rational varieties

We study finite $p$-subgroups of birational automorphism groups. By virtue of boundedness theorem of Fano varieties, we prove that there exists a constant $R(n)$ such that a rationally connected variety of dimension $n$ over an algebraically closed field is rational if its birational automorphism group contains a $p$-subgroups of maximal rank for $p > R(n)$. Some related applications on Jordan property are discussed.

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…

利用birational变换特征化变种

Characterizing Varieties Using Birational Transformations

该研究通过birational变换和基域自同构来特征化变种，揭示了被规则变种与非被规则变种在birational变换群性质上的差异。

On explicit birational geometry for weak Fano varieties and polarised Calabi-Yau varieties

Given a natural number $l$ and a weak Fano $n$-fold $X$ with $\operatorname{dim}\overline{φ_{-lK_X}(X)}\geq n-1$, we study the lower bound of the anti-canonical volume and the upper bound of the anti-canonical stability index. The method can also be used to give similar bounds for polarised Calabi-Yau varieties.

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。

Birationally rigid varieties with a pencil of Fano double covers. III

We complete the study of birational geometry of Fano fiber spaces $π\colon V\to {\mathbb P}^1$, the fiber of which is a Fano double hypersurface of index 1. For each family of these varieties we either prove birational rigidity or produce explicitly non-trivial structures of Fano fiber spaces. A new linear method of studying movable systems on Fano fiber spaces $V/{\mathbb P}^1$ is developed.

Higher index Fano varieties with finitely many birational automorphisms

Determining when the birational automorphism group of a Fano variety is finite is an interesting and difficult problem. The main technique for studying this problem is by the Noether-Fano method. This method has been effective in studying this problem for Fano varieties of index one and two. The purpose of this paper is to give a new approach to this problem, and to show that in every positive characteristic there are Fano varieties of arbitrarily large index with finite (or even trivial) birational automorphisms. To do this we prove that these varieties admit ample and birationally equivarian…

Birational aspects of the Geometry of Varieties of Sum of Powers

Varieties of Sums of Powers describe the additive decompositions of an homogeneous polynomial into powers of linear forms. Despite their long history, going back to Sylvester and Hilbert, few of them are known for special degrees and number of variables. In this paper we aim to understand a general birational behaviour of VSP, if any. To do this we birationally embed these varieties into Grassmannians and prove the rationality, unirationality or rational connectedness of many of those in arbitrary degrees and number of variables.

On the Chow ring of birational irreducible symplectic varieties

We show that the graded Chow rings of two birational irreducible symplectic varieties are isomorphic. This lifts a result known for the cohomology algebras to the level of Chow rings, despite the non-injectivity the cycle class map. In the special case of general Mukai flops, we present an alternative approach based on explicit calculations.

Birational characterization of abelian varieties and ordinary abelian varieties in characteristic p>0

Let $k$ be an algebraically closed field of characteristic $p>0$. We give a birational characterization of ordinary abelian varieties over $k$: a smooth projective variety $X$ is birational to an ordinary abelian variety if and only if $κ_S(X)=0$ and $b_1(X)=2 \dim X$. We also give a similar characterization of abelian varieties as well: a smooth projective variety $X$ is birational to an abelian variety if and only if $κ(X)=0$, and the Albanese morphism $a: X \to A$ is generically finite. Along the way, we also show that if $κ_S (X)=0$ (or if $κ(X)=0$ and $a$ is generically finite) then the A…

Birationally rigid varieties with a pencil of Fano double covers. II

We continue to study birational geometry of Fano fibrations $π\colon V\to {\mathbb P}^1$ the fibers of which are Fano double hypersurfaces of index 1. For a majority of families of this type, which do not satisfy the condition of sufficient twistedness over the base, we prove birational rigidity (in particular, it means that there are no other structures of a fibration into rationally connected varieties) and compute their groups of birational self-maps. We considerably improve the principal components of the method of maximal singularities, in the first place, the technique of counting multip…

Generic Vanishing Index and the Birationality of the Bicanonical Map of Irregular Varieties

We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety X with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibered by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.

具有有理等价 Fano 直线簇的三次四重体

Cubic fourfolds with birational Fano varieties of lines

本文给出了几对非同构三次四重体例子，证明其 Fano 直线簇是有理等价的，为 Fourier-Mukai 伙伴猜想提供新证据，并提出三次四重体与其 Fano 直线簇有理等价性相关的猜想。

关于特征 $p>0$ 下 Fano 簇的典范丛公式与有效双有理性

On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$

本文研究了正特征下 Fano 型簇的双有理几何与有界性问题，证明了有界性在正规化下不变，给出了更易用的典范丛公式，并证实了具有良好奇点的弱 Fano 簇的有效双有理性。

Secant Varieties and Birational Geometry

We show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations. These were first constructed as flips in the case of curves by M. Thaddeus via Geometric Invariant Theory, and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper. We expose the finer structure of a second flip; again for varieties of arbitrary dimension. We also prove a result on the cubic generation of the secant variety and give some conjectures o…

包含三次滚动面对的三次四重线上Fano簇的双有理几何

Birational geometry of Fano varieties of lines on cubic fourfolds containing pairs of cubic scrolls

研究包含三次滚动面对的三次四重线上Fano簇（超Kähler四重）的双有理几何，识别所有双有理模型并给出显式映射，同时刻画双有理自同构群的生成元和关系。

On explicit birational geometry for polarised varieties

In this paper, we investigate the explicit birational geometry for projective $ε$-lc varieties polarised by nef and big Weil divisors. We show that if $X$ is a projective $ε$-lc variety, $H$ is a nef and big Weil divisor with $\dim\overline{φ_{H}(X)}\geq n-1$ and $L$ is an effective Weil divisor such that $|L-K_X|\neq \emptyset$ or $L-K_X$ is nef, then we can find an explicit lower bound of $\text{vol}(H)$ and prove that $|L+m^\prime H|$ is birational for $m^\prime\geq m$, where $m$ is an explicit number which depends only on $n$ and $ε$. This result can be applied to polarised Calabi-Yau vari…

Holomorphic symmetric differentials and a birational characterization of Abelian Varieties

A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational characterization of abelian varieties. In particular we prove that, under the conjectures of the Minimal Model Program, a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections.

容许良好极小模型的光滑族的双有理几何

Birational geometry of smooth families of varieties admitting good minimal models

本文通过构造典范奇点极化簇的模函子，证明具良好极小模型的非双有理同平凡射影流形族其基空间承载正Kodaira维数的对数多微分子层，并在特殊基概形上证实Kebekus–Kovács猜想。

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。

Birational Geometry of Toric Varieties

Toric geometry provides a bridge between algebraic geometry and combinatorics of fans and polytopes. For each polarized toric variety (X,L) we have associated a polytope P. In this thesis we use this correspondence to study birational geometry for toric varieties. To this end, we address subjects such as Minimal Model Program, Mori fiber spaces, and chamber structures on the cone of effective divisors. We translate some results from these theories to the combinatorics of polytopes and use them to get structure theorems on space of polytopes. In particular, we treat toric varieties known as 2-F…

On birational boundedness of Fano fibrations

We investigate birational boundedness of Fano varieties and Fano fibrations. We establish an inductive step towards birational boundedness of Fano fibrations via conjectures related to boundedness of Fano varieties and Fano fibrations. As corollaries, we provide approaches towards birational boundedness and boundedness of anti-canonical volumes of varieties of $ε$-Fano type. Furthermore, we show birational boundedness of $3$-folds of $ε$-Fano type.

Chern classes of birational varieties

A theorem of Batyrev's asserts that if two nonsingular varieties V,W are birational, and their canonical bundles agree after pull-back to a resolution of indeterminacies of a birational map between them, then the Betti numbers of V and W coincide. We prove that, in the same hypotheses, the total homology Chern classes of V and W are push-forwards of the same class in the Chow group of the resolution. For example, it follows that the push-forward of the total Chern class of a crepant resolution of a singular variety is independent of the resolution.

Birational geometry of algebraic varieties with a pencil of Fano complete intersections

We prove birational superrigidity of generic Fano fiber spaces $V/{\mathbb P}^1$, the fibers of which are Fano complete intersections of index 1 and dimension $M$ in ${\mathbb P}^{M+k}$, provided that $M\geq 2k+1$. The proof combines the traditional quadratic techniques of the method of maximal singularities with the linear techniques based on the connectedness principle of Shokurov and Koll\' ar. Certain related results are also considered.

Birational Mori fiber structures of $\mathbb{Q}$-Fano $3$-fold weighted complete intersections, III

This is a continuation of a series of papers studying the birational Mori fiber structures of anticanonically embedded $\mathbb{Q}$-Fano $3$-fold weighted complete intersections of codimension $2$. We have proved that $19$ families consists of birationally rigid varieties and $14$ families consists of birationally birigid varieties. The aim of this paper is to continue the work systematically and prove that, among the remaining $52$ families, $21$ families consist of birationally birigid varieties.

Birational geometry of algebraic varieties, fibred into Fano double spaces

We develop the quadratic technique of proving birational rigidity of Fano-Mori fibre spaces over a higher-dimensional base. As an application, we prove birational rigidity of generic fibrations into Fano double spaces of dimension $M\geqslant 4$ and index one over a rationally connected base of dimension at most $\frac12 (M-2)(M-1)$. An estimate for the codimension of the subset of hypersurfaces of a given degree in the projective space with a positive-dimensional singular set is obtained, which is close to the optimal one.

Projective and birational geometry of Grassmannians and other special varieties

In the first part of the thesis, we study a classical invariant of projective varieties, the secant defectivity. The second part is devoted to modern algebraic geometry, we study the birational geometry of blow-ups of Grassmannians at points.

Equivalent birational embedding IV: reduced varieties

Two reduced projective schemes are said to be Cremona equivalent if there is a Cremona map that maps one in the other. In this paper I revise some of the known results about Cremona equivalence and extend the main result of [MP09] to reduced schemes. This allows to prove a very general contractibility result for union of rational subvarieties.

Birational geometry of defective varieties, II

Let $X \subset \mathbb{P}^r$ be smooth and irreducible and for $k \ge 0$ let $ν_k(X)$ (resp., $δ_k(X)$) be the $k$-th contact (resp., the $k$-th secant) defect of $X$. For all $k \ge 0$ we have the inequality $ν_k(X) \ge δ_k(X)$ and in the case $k=1$ we characterize projective varieties $X$ for which equality holds, $\dim \mathrm{Sing}(X) \le δ_1(X) -1$ and the generic tangential contact locus is reducible.

Valuation Theory, Riemann Varieties and the Structure of integral Preschemes

In this work we show that the classical subject of general valuation theory and Zariski-Riemann varieties has a much wider scope than commutative algebra and desingularization theory. We construct and investigate birational projective limit objects appropriate for the study of countably many birational models at one time. We use nonseparated Riemann varieties to investigate the birational structure of integral preschemes satisfying the existence condition of the valuative criterion of properness.

On the birational motive of hyper-Kähler varieties

We introduce a new ascending filtration, that we call the co-radical filtration in analogy with the basic theory of co-algebras, on the Chow groups of pointed smooth projective varieties. In the case of zero-cycles on projective hyper-Kähler manifolds, we conjecture it agrees with a filtration introduced by Voisin. This is established for moduli spaces of stable objects on K3 surfaces, for generalized Kummer varieties and for the Fano variety of lines on a smooth cubic fourfold. Our overall strategy is to view the birational motive of a smooth projective variety as a co-algebra object with res…

Toric Fano varieties and birational morphisms

In this paper we study smooth toric Fano varieties using primitive relations and toric Mori theory. We show that for any irreducible invariant divisor D in a toric Fano variety X, we have $0\leqρ_X-ρ_D\leq 3$, for the difference of the Picard numbers of X and D. Moreover, if $ρ_X-ρ_D>0$ (with some additional hypotheses if $ρ_X-ρ_D=1$), we give an explicit birational description of X. Using this result, we show that when dim X=5, we have $ρ_X\leq 9$. In the second part of the paper, we study equivariant birational morphisms f whose source is Fano. We give some general results, and in dimension …

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.

Piecewise morphisms of birational foliated varieties

This paper has been withdrawn by the author due to a crucial error.

Birational maps between Calabi-Yau manifolds associated to webs of quadrics

We consider two varieties associated to a web of quadrics W in the projective space of dimension 7. One is the base locus and the second one is the double cover of the three dimensional projective space branched along the determinant surface of W. We show that small resolutions of these varieties are Calabi-Yau manifolds. We compute their Betti numbers and show that they are not birational in the generic case. The main result states that if the base locus of W contains a plane then in the generic case the two varieties are birational.

Inversion maps and torus actions on rational homogeneous varieties

Complex projective algebraic varieties with $\mathbb{C}^*$-actions can be thought of as geometric counterparts of birational transformations. In this paper we describe geometrically the birational transformations associated to rational homogeneous varieties endowed with a $\mathbb{C}^*$-action with no proper isotropy subgroups.

Small bound for birational automorphism groups of algebraic varieties (with an Appendix by Yujiro Kawamata)

We give an effective upper bound of |Bir(X)| for the birational automorphism group of an irregular n-fold (with n = 3) of general type in terms of the volume V = V(X) under an ''albanese smoothness and simplicity'' condition. To be precise, |Bir(X)| < d_3 V^{10}. An optimum linear bound |Bir(X)|-1 < (1/3)(42)^3 V is obtained for those 3-folds with non-maximal albanese dimension. For all n > 2, a bound |Bir(X)| < d_n V^{10} is obtained when alb_X is generically finite, alb(X) is smooth and Alb(X) is simple.

Severi-Brauer 簇的双有理等价

Birational equivalence of Severi-Brauer varieties

本文证明了指数不是素数幂的 Severi-Brauer 簇的 Amitsur 猜想。

Birational boundedness of low dimensional elliptic Calabi-Yau varieties with a section

We prove that there are finitely many families, up to isomorphism in codimension one, of elliptic Calabi-Yau manifolds $Y\rightarrow X$ with a rational section, provided that $\dim(Y)\leq 5$ and $Y$ is not of product-type. As a consequence, we obtain that there are finitely many possibilities for the Hodge diamond of such manifolds. The result follows from log birational boundedness of klt pairs $(X, Δ)$ with $K_X+Δ$ numerically trivial and not of product-type, in dimension at most $4$.

Fano 簇的双有理刚性与 alpha 不变量

Birational rigidity and alpha invariants of Fano varieties

本文构造双有理超刚性及带有限群作用的 Fano 簇，证明其 alpha 不变量可任意接近 $1/2$，群版本 alpha 不变量可任意小。

Biregular and birational geometry of quartic double solids with 15 nodes

Three-dimensional del Pezzo varieties of degree 2 are double covers of projective space $\mathbb{P}^{3}$ branced in a quadric. In this paper we prove that if a del Pezzo variety of degree 2 has exactly 15 nodes then the corresponding quadric is a hyperplane section of the Igusa quartic or, equivalently, all such del Pezzo varieties are members of one particular linear system on the Coble fourfold. Their automorphism groups are induced from the automorphism group of Coble fourfold. Also we classify all $G$-birationally rigid varieties of such type.

Valuation theoretic methods in the birational geometry of algebraic varieties

In this paper, we give a valuation formula for rational top differential forms of function fields in characteristic zero for arbitrary Abhyankar places generalizing the classical valuation at prime divisors. This enables us to define log discrepancies for log pairs for arbitrary Abhyankar places. If the Abhyankar place has dimension greater than zero we restrict rational top differential forms with valuation zero to the residue field of the Abhyankar place, generalizing the classical restriction of a top differential form with a simple pole along a smooth divisor. This opens up the door to gen…

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the n…

原始辛簇双有理几何的脚注

Footnotes to the birational geometry of primitive symplectic varieties

本文将不可约全纯辛流形双有理几何的若干结果推广至奇异情形，补充了原始辛簇研究的相关注记。

Syzygies, Pluricanonical Maps, and the Birational Geometry of Varieties of Maximal Albanese Dimension

In this thesis we looked into three different problems which share, as a common factor, the exstensive use of the Fourier--Mukai transform as research tool. In the first Part we investigated the syzygies of Kummer varieties (i.e. quotients of abelian varieties by the $\mathbb{Z}/2\Z$-action induced by the group operation), extending to higher syzygies results on projective normality and degree of equations of Sasaki, Kempf, and Khaled. The second Part of this Thesis (partially written in collaboration with Z.Jiang and M. Lahoz) is dedicated to the study of pluricanonical linear systems on vari…

Pluricanonical maps of varieties of Albanese fiber dimension two

In this paper we prove that for any smooth projective variety of Albanese fiber dimension two and of general type, the 6-canonical map is birational. And we also show that the 5-canonical map is birational for any such variety with some geometric restrictions.

大类上K稳定性的一种双有理版本

A birational version of K-stability for big classes

本文提出大线丛上光滑射影簇的统一K稳定性理论，扩展了已有理论，并证明统一K稳定性在拉回和某些推前下保持，从而在b除子层面定义明确。

Finite groups of birational transformations

We survey new results on finite groups of birational transformations of algebraic varieties.

Birational geometry of cluster algebras

We give a geometric interpretation of cluster varieties in terms of blowups of toric varieties. This enables us to provide, among other results, an elementary geometric proof of the Laurent phenomenon for cluster algebras (of geometric type), extend Speyer's example of an upper cluster algebra which is not finitely generated, and show that the Fock-Goncharov dual basis conjecture is usually false.

$K$-平凡簇的动机zeta函数的双有理不变性，以及光滑填充的障碍

Birational invariance of motivic zeta functions of $K$-trivial varieties, and obstructions to smooth fillings

本研究证明$K$-平凡簇的动机zeta函数及相关单值性猜想具有双有理不变性，扩展了已知适用范围，并揭示该函数可作为簇在$t=0$处光滑填充存在性的障碍。

Specialization of birational types

We introduce a simplified version of the Grothendieck group of algebraic varieties and use it to show that birational types specialize in families with mild singularities of the central fiber.

Birational models for varieties of Poncelet curves

We consider Poncelet pairs $(S,C)$, where $S$ is a smooth conic and $C$ is a degree$-c$ plane curve having the Poncelet property with respect to $S$. We prove that for $c>4$ the projection $(S,C)\mapsto C$ is generically one-to-one and use this to describe a birational model of the variety of Poncelet curves for $c$ odd.

Finite groups of birational selfmaps of threefolds

We determine three-dimensional algebraic varieties whose groups of birational selfmaps do not satisfy the Jordan property.