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DTSTART;TZID=Europe/Berlin:20260619T133000
DTEND;TZID=Europe/Berlin:20260619T143000
DTSTAMP:20260619T020031
CREATED:20260519T104110Z
LAST-MODIFIED:20260519T104110Z
UID:13427-1781875800-1781879400@crc326gaus.de
SUMMARY:On arithmetical surjectivity and the Conjecture of Colliot-Thelene
DESCRIPTION:Florian Pop (University of Pennsylvania) \nThe notion of ‘arithmetical surjectivity’ (a.s.) for dominant morphisms f of proper smooth varieties over number fields was introduced by Colliot-Thelene\, and he made a precise\nconjecture (CCT) relating a.s. to birational properties of the morphisms f. The CCT was proved in a sharper form by Denef (2019)\, and Loughran-Skorobogatov-Smeets gave a\ncharacterization of a.s. (2020). I will present a new method of proof which allows generalizations/refinements of the above results by: First\, allowing k to be any finitely generated base fields k with char(k)=0 (and beyond). Second\, showing that a.s. is a fully birational property\, i.e.\, a.s. depends only on properties of the function field extension defined by morphisms f. The method of proof also yields generalizations of the so called\nzero-cycle surjectivity\, considered/characterized over number fields by Gvirtz (2020).\nNOTE: The problems are completely open in positive characteristic!
URL:https://crc326gaus.de/event/on-arithmetical-surjectivity-and-the-conjecture-of-colliot-thelene/
LOCATION:Heidelberg\, Mathematikon\, SR A and Livestream
CATEGORIES:GAUS-Seminar
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DTSTART;TZID=Europe/Berlin:20260619T153000
DTEND;TZID=Europe/Berlin:20260619T170000
DTSTAMP:20260619T020031
CREATED:20260319T100422Z
LAST-MODIFIED:20260520T110437Z
UID:12879-1781883000-1781888400@crc326gaus.de
SUMMARY:Seminar on Arithmetic Geometry
DESCRIPTION:Louisa Bröring (Duisburg-Essen): Quadratic Euler Characteristic of Geometrically Cyclic Branched Coverings \nThe quadratic Euler characteristic $\chi(X)$ of a smooth\, projective scheme\n$X$ over a field $k$ of characteristic not two is a refinement of the\ntopological Euler characteristic to quadratic forms\, constructed using motivic\nhomotopy theory. For example\, if $k\subset \mathbb{R}$\, then rank of $\chi(X)$\nis equal to the topological Euler characteristic of $X(\mathbb{C})$ and the\nsignature of $\chi(X)$ with respect to the given embedding is equal to the\ntopological Euler characteristic of $X(\mathbb{R})$. The quadratic Euler\ncharacteristic plays an import role in the programme of $\mathbb{A}^1$-refined\nenumerative geometry. \nAfter briefly introducing the quadratic Euler characteristic\, we present a\ncomputation of the quadratic Euler characteristic of geometrically cyclic\nbranched coverings leveraging Levine’s quadratic Riemann-Hurwitz formula. An\n$n$-fold geometrically cyclic branched covering is a morphism $f\colon Y \to\nX$ between smooth\, projective schemes together with a smooth\, closed subscheme\n$Z \subset X$ satisfying the following condition: there exists a line bundle\n$L$ over $X$ and a section $s \colon X \to L^{\otimes n}$ such that $Z$ is the\nzero locus of $s$ and $f$ is the pullback along $s$ of the map $L \to\nL^{\otimes n}$ taking $n$-th powers. \nAs an application\, we compute the quadratic Euler characteristic of branched\ndouble covers of $\mathbb{P}^2$\, which includes some K3 surfaces. \nZoom (635 7328 0984\, Kenncode: kleinste sechsstellige Primzahl)
URL:https://crc326gaus.de/event/seminar-on-arithmetic-geometry-35/
LOCATION:Darmstadt\, Room 401 and Zoom\, Schlossgartenstraße 7\, Darmstadt\, 64289\, Germany
CATEGORIES:GAUS-Seminar
ORGANIZER;CN="Sabrina Pauli":MAILTO:pauli@mathematik.tu-darmstadt.de
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