HIGHER DIMENSIONAL ARITHMETIC GEOMETRY:PROGRAM• Feb 18th (Mon)– 9:30 - 10:30 Tim Browning– 11:00 - 12:00 Fumiharu Kato
– 12:00 - 14:00 Lunch– 14:00 - 15:00 Marta Pieropan– 15:30 - 16:30 Yusuke Nakamura• Feb 19th (Tue)– 9:30 - 10:30 Takehiko Yasuda– 11:00 - 12:00 Damaris Schindler– 12:00 - 14:00 Lunch– 14:00 - 15:00 Yu Yasufuku– 15:30 - 16:30 Daniel Loughran• Feb 20th (Wed)– 9:00 - 10:00 Cecı́lia Salgado– 10:30 - 11:30 Keiji Oguiso– 11:30 - 13:30 Lunch– 13:30 - 14:30 Arne Smeets– 15:00 - 16:00 Brian Lehmann– 16:30 - 17:30 Yohsuke Matsuzawa• Feb 21st (Thu)– 9:30 - 10:30 Yoshinori Gongyo– 11:00 - 12:00 Alexei Skorobogatov– 12:00 - 14:00 Lunch

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Titles and Abstracts• Tim Browning
Title: A mixed Waring problem and its application to higher-dimensional
Campana orbifolds
Abstract: Waring’s problem concerns how many ways a positive integer canbe represented as a sum of powers. Using the recent resolution of the main
conjecture in Vinogradov’s mean value theorem (due to Bourgain-DemeterGuth and Wooley), we can study this problem in considerable generality
and apply it to estimate the density of rational points on higher-dimensional
orbifolds (Pn , D) when D is a Q-divisor involving hyperplanes. This is joint
work with Shuntaro Yamagishi.
• Fumiharu Kato
Title: Non-archimedean analytic space of Banach type (Joint-work with K.
Fujiwara)
Abstract: In our book ”Foundations of rigid geometry I” (from EMS) withKazuhiro Fujiwara, we have introduced what we called R+ -metrized analytic space, of which the affinoid patches are valuative spectra of R+ -filteredvaluations. The category of these spaces is a crossroad of Huber’s adicspaces and Berkovich spaces. I will describe in the talk the construction ofthese spaces, and discuss further properties of them.
• Marta Pieropan
Title: On rationally connected varieties over C1 fields of characteristic 0
Abstract: In the 1950s Lang studied the properties of C1 fields, that is, fields
over which every hypersurface of degree at most n in a projective space ofdimension n has a rational point. Later he conjectured that every smoothproper rationally connected variety over a C1 field has a rational point. The
conjecture is proven for finite fields (Esnault) and function fields of curves
over algebraically closed fields (Graber-Harris-de Jong-Starr), but it is still
open for the maximal unramified extensions of p-adic fields. I use birational
geometry in characteristic 0 to reduce the conjecture to the problem offinding rational points on Fano varieties with terminal singularities.
• Yusuke NakamuraTitle: A rational point problem on Fano varieties
Abstract: Fano varieties defined over a certain field k are believed to havea k-rational point. For instance, the classicallyknown Chevalley–Warningtheorem states that homogeneous polynomials f1 , . . . , fl ∈ Fq [x0 , . . . , xn ]with (n + 1)-many
variables over a finite field Fq have a non-trivial commonP
solution if
1≤i≤l deg fi ≤ n holds. More precisely, it states that thenumber of common solution is divisible by the characteristic p of the field
Fq . Geometrically, it can be interpreted as that the number of the Fq rational points on a complete intersection Fano variety is one modulo p.
This kind of problem was answered in the smooth case by Esnault. She
proved that smooth Fano varieties defined over a finite field have a rational
point. It is natural to ask whether we can generalize this results to singularHIGHER DIMENSIONAL ARITHMETIC GEOMETRY: PROGRAM

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Fano varieties or not. In this talk, we study the rational point problem on
singular Fano varieties and the relation to the minimal model program.
• Takehiko YasudaTitle: Computational aspects of the McKay correspondence
Abstract: I will talk about the following (mainly computational) aspectsof the (tame or wild) McKay correspondence: (1) the tame McKay correspondence over a finite field (joint work with Melanie Wood), (2) nonlinear actions and rational double points and (3) pathological examples byYamamoto.
• Damaris Schindler
Title: Counting points close to varieties
Abstract: We discuss small solutions to ternary diagonal inequalities of any
degree where all of the variables are assumed to be of size P . We study
this problem on average over a one-parameter family of forms and discussa generalization of work of Bourgain on generic ternary diagonal quadratic
forms to higher degree. In particular we discuss how these Diophantine
inequalities are related to counting rational points close to varieties.
• Yu Yasufuku
Title: Height inequalities from cohomological invariants
Abstract: Recently, McKinnon–Roth introduced a constant (related to theSeshadri constant) measuring how well an algebraic point approximates
an ample line bundle, and obtained Schmidt-type height inequalities. Ru–Vojta also introduced a related constant using asymptotic of the numberof global sections, and applied to both arithmetic setting and Nevanlinnatheoretic setting. Ru–Wang then generalized to approximating by subvarieties. Here we apply these theories to exceptional divisors, and discuss its
connection with the gcd results of Levin.
• Dan Loughran
Title: Integral points on stacks
Abstract: In this talk we discuss analogues for stacks of various classicaltheorems in arithmetic geometry (Hermite-Minkowski and Chevalley-Weil).
This is joint work with Ariyan Javanpeykar.
• Cecı́lia Salgado
Title: Variation of the Mordell-Weil rank in families of abelian varieties
Abstract: We consider a family of abelian varieties over a number field K ,i.e. a variety X with a map to a curve B whose fibres are abelian varieties.
The generic fiber is an abelian variety over the function field K(B) and thegroup of K(B)-rational points has a rank r. For almost all point t in B(K)the fiber is an abelian variety Xt over K and the group of K-rational point
has rank r(t). A specialization theorem of Silverman states that for almost
all point t in B(K) the rank r(t) is greater or equal to r. In a joint workwith M. Hindry we show that, under certain geometric conditions, there areinfinitely many t’s with r(t) > r. In a work in progress with D. Loughran,we address the question of whether this infinite set of rank jump is thin or

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not. I will report on both projects and provide some interesting examples
on which we achieve a larger rank jump within fibers.
• Keiji OguisoTitle: A surface in odd characteristic with discrete and non-finitely generated automorphism group
Abstract: It was proved by Professor Tien-Cuong Dinh and me that there isa smooth complex projective surface whose automorphism group is discreteand not finitely generated. In this talk, I would like to show that there isa smooth projective surface, birational to some K3 surface, such that theautomorphism group is discrete and not finitely generated, over any algebraically closed field of odd characteristic except precisely an algebraic
closure of the prime field.
• Arne Smeets
Title: Orbifold Mordell over function fields
Abstract: I will introduce Campana’s ”orbifold pairs”, in particular the1-dimensional version, and explain how to formulate a version of Mordell’s
conjecture for such pairs. Over number fields, proving such a statement
seems out of reach: all we know is that it would follow from the abc conjecture. Over function fields however, such a result can actually be proven,both in characteristic zero and in positive characteristic; I will give anoverview of the proof and the techniques involved. (Joint work with Stefan
Kebekus and Jorge Vitorio Pereira.)• Brian Lehmann
Title: Sections of del Pezzo fibrations over P1
Abstract: I will describe a framework for classifying sections of a del Pezzo
surface fibration over P1 . Our approach is motivated by the function fieldversion of Manin’s Conjecture, and I will explain the relationship between
the two. I will also mention some applications to Gromov-Witten invariants
and to the study of the Abel-Jacobi map. This is joint work with ShoTanimoto.
• Yohsuke Matsuzawa
Title: An approach to Kawaguchi-Silverman conjecture for self-morphismsof projective varieties
Abstract: In the study of arithmetic dynamics over number fields, it is important to understand the growth rate of Weil height functions along theorbits of rational self-maps. Arithmetic degree is a measure of the growthrate and Kawaguchi-Silverman conjecture asserts that the arithmetic degree of every Zariski dense orbit is equal to the first dynamical degree ofthe self-map. I will focus on the case where the self-map is a morphism anddiscuss an approach to this conjecture using equivariant MMP.
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5• Yoshinori Gongyo
Title: My understanding of proofs of BAB conjecture and ACC conjecture
for LCT.
Abstract: I wil explain the inductive argument for the proof of ACC conjecture for LCT and the BAB conjecture after Birkar and Hacon–Mckernan–
Xu. Historically the proof of the ACC was done before that of the BAB.
This was involved with the DCC property of volumes of log canonical bundle of log general types. At this time, I will give proofs of the both in thecategory of log Fano varieties. So we do not involve the DCC property ofvolumes of log canonical bundle of log general types. This is some kind ofone tower proof of ACC for LCT and the BAB. I do not think that I cangive all of proofs but I will pick some reason why the inductive arguments
work.
• Alexei Skorobogatov
Title: On the Brauer group of diagonal surfaces
Abstract: Using work of Pham and Looijenga on the complex cohomologyof Fermat varieties, and work of Weil, Katz, Shioda and Ulmer on theassociated Galois representation, we propose a framework for calculating
the Brauer group of a surface given by a diagonal equation over a numberfield. In the case of quartic diagonal surfaces with rational coefficients wedetermine the Brauer group modulo the Brauer group of the ground field
k, in the case when k = Q or k = Q(i). This is a joint work with DaminGvirtz