Special SFB-seminar day - Supersingular K3 surfaces are unirational (introduction)

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 (Shioda-supersingularity), the formal Brauer group has infinite height (Artin-supersingularity), 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.