Whichvarieties are birational to Ph rational Difficult problem k x k - PowerPoint PPT Presentation

Irrationality of quotient varieties Urban Jezernik University of the Basque Country Funded by EU Horizon 2020 under MScgrant 748129

Whichvarieties are birational to Ph rational Difficult problem k x k ti Tn smooth projective variety X rk Problem Given a nice is X birational to some Ph

Whichvarieties are birational to Ph rational Difficult problem k x k ti Tn smooth projective variety X rk Problem Given a nice is X birational to some Ph Liiroth problem unirational nice g Ip teen kCX X kfta th when dim X E Z iiroth problem has a positiveanswer over k fi

Noether problem quotient variety nice G IG Spec KEV f vectorspace representation of G G invariant regular functions finite group IAI to C x Cz A V x y Example y E Speck U v wJ Spec k x2 ay VIC Cuw v2 y E k Cee v k Vce P2 Vez

Motivation If VG is birational to Ph over k Q then one can strongly solve the inverse Gabi's problem for G o G E Gall EIQ for some E Noether Hilbert

Motivation If VG is birational to IP k Q then one over can strongly solve the inverse Gabi's problem for G o G E Gall EIQ for some E Noether Hilbert Noetherproblem does not have a positive answer Unfortunately smallest counterexample is Cg in general Lenstra 1974 Plans 2018 Swan 1969

Motivation If VG is birational to IP k Q then one over can strongly solve the inverse Gabi's problem for G o G E Gall EIQ for some E Noether Hilbert Noetherproblem does not have a positive answer Unfortunately smallest counterexample is Cg in general Lenstra 1974 Plans 2018 Swan 1969 Assume from now on that k E

No namelemma The question of whether or not WG is stably rationaldoes not depend on the representation YG P is rational for some h C Miyata1971

No namelemma The question of whether or not WG is stably rationaldoes not depend on the representation YG P is rational for some h C Miyata1971 Example KG rational Fischer 1915 G abelian

No namelemma The question of whether or not WG is stably rationaldoes not depend on the representation YG P is rational for some h C Miyata1971 Example KG rational Fischer 1915 G abelian KG rational G Sym n V G symmetric polynomials

No namelemma The question of whether or not WG is stably rationaldoes not depend on the representation YG P is rational for some h C Miyata1971 Example KG rational Fischer 1915 G abelian KG rational G Sym n V G symmetric polynomials VG rational 1Gt I 32 G 2 group Chu Hui Kang Prokhorov 2008

To prove that a variety X is not stably rational one uses I with a birational invariant abelian groups varieties I X f 0 I Ph O

To prove that a variety X is not stably rational one uses I with a birational invariant abelian groups varieties I X f 0 I Ph O Artin Torsion Hfing X Mumford 1971 E Z

To prove that a variety X is not stably rational one uses I with a birational invariant abelian groups varieties I X f 0 I Ph O Artin Torsion Hfing X Mumford 1971 E Z N Br Ec x Br Brnr X R image R DVR Salt man 1984 for X VIG Ek KCR unramified Brauer group

To prove that a variety X is not stably rational one uses I with a birational invariant abelian groups varieties I X f 0 I Ph O Artin Torsion Hfing X Mumford 1971 E Z N Br Ec x Br Brnr X R image R DVR Salt man 1984 for X VIG Ek KCR unramified Brauer group For X smooth projective unirational Z Torsion H3sing X E Brw X Br X Bogomolov 1989

This invariant is computable for X VIG A kernel H4G HEA ex Brar VIG AE G Bogomolov 1989 A abelian Bogomolov multiplier BoCG

This invariant is computable for X VIG A kernel H4G HEA ex Brar VIG A EG Bogomolov 1989 A abelian th NG Bogomolov multiplier BoCG Higher unramified cohomology Colliot The leine Ojanguren 1989 id Him X EE't in Leray spectral sequence Xan Xzar Only partial extensions of Bogomolov's formula for X VIG Hoshi Kanga Yamasaki 2018 computer calculations Peyre2008

Example Bo abeliangroup O

Example Bo abeliangroup O Bo finite simple group 0 Kunyavski 2010

Example Bo abeliangroup O Bo finite simple group 0 Kunyavski 2010 Bo exponent p class 2 quotient of Isl 2g for g 2 2 O g Saltman 1984 ITExi y 1 i n

Example Bo abeliangroup O Bo finite simple group 0 Kunyavski 2010 Bo exponent p class 2 quotient of Isl 2g for g 22 O g Saltman 1984 ITExi y 1 i n Chul B some groups of order 64 0 Hu Kang Kunyarski 2012 Hoshi Kang KunyavskT 2012 Bo some groups of order p5 1 0

Quite surprisingly Bo is often non trivial IG I p h lBoc G I 2 M log G AM him 1 a log IG I G p with Sanchez 2018 It follows that Noether problem has a strongly negative answer

an attractive object Bo is

an attractive object reappears in Bo is it K theory B G kernel Ky 2pG SKI Epa Kr Qp G Oliver 1980

an attractive object reappears in Bo is it K theory B G kernel Ky 2pG SKI Epa Kr Qp G Oliver 1980 Representation theory 1 I B G 1 1 I G ab l 1 p 11 11 Lin VC Lin I pg FpG with Garcia Rodriguez Kin Zapirain 2017 Jai

Mathematical physics soft tensor braided auto X Bo G equivalences Outage G of Drinfeldcente.ro category of compatible Davydov 2014 G graded G vector spaces

Mathematical physics soft tensor braided auto X Bo G equivalences Outage G of DrinfeldcenterofCI category of compatible Davydov 2014 G graded G vector spaces Moduli spaces Hurwitzspace of connected curves G covering D B aepapihantngmesu.FI I e aymmongdro mieses Ellenberg Venkatesh Westerland 2012