Một số vấn đề về đa tạp fano chiều thấp

MINISTRY OF EDUCATION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND TECHNOLOGY Institute of Mathematics NGUYỄN THỊ ÁNH HẰNG MỘT SỐ VẤN ĐỀ VỀ ĐA TẠP FANO CHIỀU THẤP (ON SOME PROBLEMS OF FANO VARIETIES IN SMALL DIMENSIONS) THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2025 MINISTRY OF EDUCATION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND TECHNOLOGY Institute of Mathematics NGUYỄN THỊ ÁNH HẰNG MỘT SỐ VẤN ĐỀ VỀ ĐA TẠP FANO CHIỀU THẤP (ON SOME PROBLEMS OF FANO VARIETIES IN SMALL DIMENSIONS) Subject area: Algebra and Number theory Subject code: 9 46 01 04 THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisor: Assoc. Hoàng Lê Trường HANOI - 2025 Abstract A complex projective algebraic variety X is said to be Am -cylindrical if there exists a principal Zariski open subset U of X such that U ∼= Z × Am where Z is an affine variety. The cylindricity of projective varieties is closely related to the flexibility of the affine cones over projective varieties. Recall that an affine variety V is called flexible if the tangent space of V at any smooth point is spanned by tangent vectors to the orbits of one- parameter unipotent subgroups of the automorphism group Aut(V ).

In this thesis, we focus on two problems. Firstly, we investigate the cylindricity of some families of smooth Fano fourfolds. Secondly, we study the flexibility of the affine cones over smooth Fano fourfolds. The dissertation consists of three chapters.

In Chapter 1, we present basic concepts related to the main results of the dissertation. In Chapter 2, which is based on the joint work with Michael Hoff and Hoang Le Truong, we focus on studying the cylindricity problem. We explore the cylindricity of three certain families of Mukai fourfolds of genus 6, 7, 9. In addition, the cylindricity of every Mukai fourfold of genus 8 is confirmed.

Note that no results on the cylindricity of Mukai fourfolds of genus 6 have been known before. In Chapter 3, which is based on the joint work with H. Truong, we focus on the flexibility problem. We explore the flexibility of the affine cone over every Mukai fourfold of genus 7.

i Declaration This dissertation was based on my research works carried out at Institute of Mathematics, Vietnam Academy of Science and Technology under the guidance of Assoc. Hoang Le Truong. All the presented results have never been published by others. Hanoi, July 2024 The author Nguyen Thi Anh Hang ii Acknowledgements First and foremost I wish to express my deep gratitude to my super- visor Associate Professor Hoang Le Truong for all the time, ideas, advice, encouragement, and guidance he has provided over the years.

He has guided and taught me with the most patience and dedication. All that support has created great motivation for me to study. I am deeply grateful to Professor Le Thi Thanh Nhan for her inspiring lectures and efforts to create a good academic environment at Thai Nguyen University. She has always cared about, helped, and encouraged me during my studies.

I sincerely thank Associate Professor Doan Trung Cuong and As- sociate Professor Tran Giang Nam. They have given me engaging lec- tures with the most enthusiasm as well as kindness and the opportunity to participate in seminars, conferences, and schools at Hanoi Institute of Mathematics during my studies. I sincerely thank Associate Professor Alexander Perepechko, HSE University, Russia for carefully reading and giving comments for the dis- sertation. I would like to thank International Mathematical Union, Hanoi Insti- tute of Mathematics and the International Center for Research and Post- graduate Training in Mathematics under the auspices of UNESCO for their support and for giving me the best condition of study so that I could com- iii plete this dissertation.

I sincerely thank Thai Nguyen University of Education, Faculty of Mathematics for giving me the most favorable conditions to both complete my studies and ensure the teaching at my host university. I also would like to thank my colleagues, PhD students at Thai Nguyen University and Hanoi Institute of Mathematics for their accom- pany, encouragement, and sharing with me not only in research but also in life. I sincerely thank all the staff of Hanoi Institute of Mathematics for their support which has enabled me to complete this work within the ex- pected time frame. Finally, I would especially like to thank my family, especially my hus- band and my two beloved children for their endless love and unconditional support.

They are a great motivation for me to try my best in study and life. The author Nguyen Thi Anh Hang iv Contents Notations. Definition of the Grassmannian. Divisors and intersection theory.

Definition of Fano varieties. On Mori contractions. Cylindricity of Mukai fourfolds of genus 6, 7, 8, 9. Cylinders in projective varieties.

Mukai fourfolds of genus 6 .1 Birational maps on Gushel-Mukai fourfolds .2 Cylindrical Mukai fourfolds of genus 6. Mukai fourfolds of genus 7 .1 Birational maps on Mukai fourfolds of genus 7 .2 Cylindrical Mukai fourfolds of genus 7. Mukai fourfolds of genus 8 .1 Birational maps on Mukai fourfolds of genus 8 .2 Cylindrical Mukai fourfolds of genus 8. Mukai fourfolds of genus 9 .1 Birational maps on Mukai fourfolds of genus 9 .2 Cylindrical Mukai fourfolds of genus 9.

Flexibility of the affine cone over Mukai fourfolds of genus 7. Birational maps on Mukai fourfolds of genus 7. 89 List of author’s related publications. 92 Notations C the field of complex numbers Ga the additive group of C R+ the set of nonnegative real numbers An the affine space of dimension n Pn the projective space of dimension n Vn V the n-th exterior power of V P(V) the projectivization of a vector space V Spec(R) the set of prime ideals of R OX the structure sheaf of X OX,x the local ring of X at the point x OX (m) the twisted sheaf ΩX the sheaf of relative differential of X ωX the canonical sheaf of X KX the canonical divisor of X −KX the anticanonical divisor of X HX the hyperplane class of X dX the degree of X dim X the dimension of X as a variety codimX Y the codimension of Y in X NY /X the normal bundle of Y in X πX = 21 HXdim X−1 ((dim X − 1)HX + KX ) + 1 1 the sectional genus of X Tp X the tangent space of X at p Pic(X) the Picard group of X ρ(X) the Picard rank of X H i (X, F) the i-th cohomology group of a sheaf F on X H i (X, Z) the i-th cohomology group of the constant sheaf with values in Z H p,q (X) ∼ p = H q (X, Ω ) X χ(F) = (−1)i dim H i (X, F) P i the Euler-Poincaré characteristic Supp(D) the support of a divisor D |D| the linear system OX (D) the sheaf associated to a divisor D ci (ε) the i-th Chern class c(ε) the total Chern class D1 ∼ D2 divisor D1 is linearly equivalent to divisor D2 D1 · D2 the intersection number f ∗D the pullback of D X2.2 complete intersection of two quadrics Q4 quadric hypersurface of dimension 4 in P5 99K birational map rankZ M the rank of a module M 2 List of Pictures Figure 2.1 : A quadric surface in A3 page 33 List of Tables Table 1.1 : Possible values of genera of Mukai varieties page 20 Table 3.1 : Surfaces Σ of degree dΣ and sectional genus πΣ page 74 contained in a Mukai fourfold X of genus 7 and surfaces F of degree dF and sectional genus πF contained in Y 3 Introduction Throughout this thesis, all varieties are assumed to be algebraic and defined over the complex number field C.

Let X be a smooth projective variety. An Am -cylinder in X is a pair (Z, φ) where Z is an affine variety and φ : Z × Am −→ X is an open embedding such that φ(Z × Am ) is a principal open subset of X. The variety X is called Am -cylindrical if there exists an Am -cylinder in X. If we have an A1 -cylinder in X, X is said to be cylindrical.

Such cylinders appear naturally in many recent problems and questions related to algebraic varieties (see [13, 14, 15, 49, 50, 76, 77, 78], Chapter 2, and Chapter 3). Studying the cylindricity of projective varieties gives us im- portant information about them. For certain smooth projective varieties, it has been proven that the cylindricity of the varieties implies rationality (see [88, 12]). Moreover, the existence of cylinders in smooth projective varieties is closely related to unipotent group actions on the affine cones over them (see [52, 25]).

Therefore, it is very important to find a complete classification of cylindrical projective varieties. The first aim of this thesis is to investigate the existence of cylinders in some families of smooth Fano varieties. A Fano variety, introduced by Gino Fano in [90, 91], is defined to be a smooth complex algebraic variety with ample anti-canonical divisor −K (see Section 1. Such varieties have been proven to play a vital role in 4 the classification of algebraic varieties and have been intensively studied in recent years (see for examples [40, 41, 42, 43, 66, 64, 65, 67, 68, 76, 77, 71]).

However, the classification of such varieties is still far from being complete. In dimension 1, the only Fano curve is the projective line P1 which is cylindrical. In dimension 2, a smooth Fano variety of dimension 2 is called a del Pezzo surface. Since every smooth rational projective surface contains a cylinder (see [12, Section 2.13]), all del Pezzo surfaces are cylindrical.

Therefore, in this case, the cylindricity problem has been fully solved. Now let X be a smooth Fano variety of dimension n ≥ 3 with Picard rank 1 and H the ample divisor generating the Picard group Pic(X) = ZH. The number i such that −KX ∼ iH is called the index of X. The index i = n + 1 if and only if X ∼ = Pn ; while i = n if and only if X is a quadric in Pn+1 (see [44, Corollary 3.

If i = n − 1, then X is called a del Pezzo variety. If i = n − 2, X is called a Fano-Mukai variety (or simply Mukai variety). In this case, the number g = 12 H n + 1 is integral and is called the genus of X. In dimension 3, it has been shown in [76, Remark 1.2] that Fano threefolds with cylinders must be rational, that is, birational to a projective space.

Although Fano threefolds are almost fully classified and well studied (see [40, 41, 42, 43, 66, 64, 65, 67, 44]), few cylindrical Fano threefolds are known even for Picard rank 1. In particular, certain Mukai threefolds of genus 9, 10 are cylindrical (see [50]). And the same is true for Mukai threefolds of genus 12 (see [51]). Some other classes of cylindrical Fano threefolds are described in [51, 12].

In dimension 4, it has been shown in [12, Corollary 1.6] that every smooth cylindrical Fano fourfold of Picard rank 1 is rational, but few ex- amples are known. Let us sum up what is known in this case. In [76], the 5 authors have shown that every smooth intersection of two quadrics X2,2 in P6 is cylindrical. Moreover, the same holds for every smooth quintic del Pezzo fourfold X5 ⊂ P7 (see [76]).

Starting with the two above-mentioned cylindrical Fano fourfolds, the authors of the papers [76, 77] have con- structed four families of cylindrical Mukai fourfolds of genus g = 7, 8, 9 and 10.