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CATEGORIES:Algebraic Geometry Seminar
SUMMARY:(COW) Standard Models of Low Degree del Pezzo Fibr
ations via GIT for Hilbert Points - Maksym Fedorch
uk\, Boston College
DTSTART;TZID=Europe/London:20190207T170000
DTEND;TZID=Europe/London:20190207T180000
UID:TALK119572AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/119572
DESCRIPTION:A del Pezzo fibration is one of the natural output
s of the Minimal Model Program for threefolds. At
the same time\, geometry of an arbitrary del Pezzo
fibration can be unsatisfying due to the presence
of non-integral fibers and terminal singularities
of an arbitrarily large index. In 1996\, Corti de
veloped a program of constructing 'standard models
' of del Pezzo fibrations within a fixed birationa
l equivalence class. Standard models enjoy a varie
ty of desired properties\, one of which is that al
l of their fibers are Q-Gorenstein integral del Pe
zzo surfaces. Corti proved the existence of standa
rd models for del Pezzo fibrations of degree d ≥ 2
\, with the case of d = 2 being the most difficult
. The case of d = 1 remained a conjecture. In 1997
\, Kollár recast and improved Corti’s result in de
gree d = 3 using ideas from the Geometric Invarian
t Theory for cubic surfaces. I will present a gene
ralization of Kollár’s approach in which we develo
p notions of stability for families of low degree
(d ≤ 2) del Pezzo fibrations in terms of their Hil
bert points (i.e.\, low degree equations cutting o
ut del Pezzos). A correct choice of stability and
a bit of enumerative geometry then leads to (very
good) standard models in the sense of Corti. This
is a joint work with Hamid Ahmadinezhad and Igor K
rylov.
LOCATION:CMS MR9
CONTACT:Mark Gross
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