Special SFBseminar day  Supersingular K3 surfaces are unirational (introduction)
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SFBSeminar
Christian Liedtke (München)
What 


When 
Dec 19, 2013 from 10:00 am to 12:00 pm 
Where  Essen, WSCNU4.03 
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Abstract:
Arguably, the most "simple" varieties are projective space
and rational varieties, i.e., varieties that are birational to
projective space. Moreover, if there exists a rational and
dominant map from projective space onto a variety, it is
called unirational, and one might hope that unirational
varieties are "close" to rational varieties. In fact, as shown
by Lüroth, a curve is the projective line if and only if it is
rational if and only if it is unirational. Also, in
characteristic zero, a surface is rational if and only if it
is unirational as shown by Castelnuovo. However, there do
exist unirational surfaces in positive characteristic that are
"very far" from being rational. Now, to understand what makes
a surface unirational, we will see that unirationality implies
that the variety in question is supersingular: algebraic
cycles fill up second cohomology (Shiodasupersingularity),
the formal Brauer group has infinite height
(Artinsupersingularity), and second crystalline cohomology
has slope one. Now, the question is: does supersingularity
conversely imply unirationality? In this talk, I will give an
introduction to the above mentioned circle of ideas, and
define all the notions mentioned.