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Oberseminar SoSe 2012 Zusammenfassung des Vortrags von Renate Vistorin

Geometry of a special K3 surface

We consider a double covering of P2 branched along a sextic, given by the equation

y2 = x0(x05 + x15 + x25).

After blowing up the singularities we get a K3 surface S. The focus of this talk will be to look for divisors that span the Picard group S. These divisors are explicitly given by two equations. We can compute intersection numbers and set up an intersection matrix. The rank of this matrix gives us a lower bound for the Picard number of S. We also show that the canonical divisor of S is trivial by computing the zeros and poles of the differential form dx1 ∧ dx2 on S.