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SUMMARY:(COW) Standard Models of Low Degree del Pezzo Fibrations via GIT f
 or Hilbert Points - Maksym Fedorchuk\, Boston College
DTSTART:20190207T170000Z
DTEND:20190207T180000Z
UID:TALK119572@talks.cam.ac.uk
CONTACT:Mark Gross
DESCRIPTION:A del Pezzo fibration is one of the natural outputs of the Min
 imal Model Program for threefolds. At the same time\, geometry of an arbit
 rary del Pezzo fibration can be unsatisfying due to the presence of non-in
 tegral fibers and terminal singularities of an arbitrarily large index. In
  1996\, Corti developed a program of constructing 'standard models' of del
  Pezzo fibrations within a fixed birational equivalence class. Standard mo
 dels enjoy a variety of desired properties\, one of which is that all of t
 heir fibers are Q-Gorenstein integral del Pezzo surfaces. Corti proved the
  existence of standard models for del Pezzo fibrations of degree d ≥ 2\,
  with the case of d = 2 being the most difficult. The case of d = 1 remain
 ed a conjecture. In 1997\, Kollár recast and improved Corti’s result in
  degree d = 3 using ideas from the Geometric Invariant Theory for cubic su
 rfaces. I will present a generalization of Kollár’s approach in which w
 e develop notions of stability for families of low degree (d ≤ 2) del Pe
 zzo fibrations in terms of their Hilbert points (i.e.\, low degree equatio
 ns cutting out del Pezzos). A correct choice of stability and a bit of enu
 merative geometry then leads to (very good) standard models in the sense o
 f Corti. This is a joint work with Hamid Ahmadinezhad and Igor Krylov.
LOCATION:CMS MR9
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