Examples of Type IV unprojection

Abstract

I show that the weighted projective line PP(2,3) has an embedding PP(2,3) into PP(4,5,6,9) whose image Gamma is contained in a quasismooth K3 hypersurface X_24 in PP(4,5,6,9). The pair (Gamma in X_24) unprojects to the codimension 4 K3 surface Y in PP(4,5,5,6,7,8,9) with Basket = [1/2(1,1), 1/5(1,4), 1/5(2,3), 1/9(4,5)], Hilbert Numerator = 1 - t^12 - t^13 - 2t^14 - 2t^15 - 2t^16 - t^17 + t^19 + 2t^20 + 3t^21 + 4t^2 + 3t^23 + .. (this example is called Altinok4(111) in the Magma K3 database). The local coordinates at the third centre P_3 = 1/5(2,3) of Y are of weight 7 and 8 (rather than 2 and 3), so both are eliminated by the projection from P_3. Together with other examples, this gives substance to Type IV unprojections. Several more cases of Type IV unprojections are known up to codimension 5 or 6. The paper also contains some Magma programming routines suitable as exercises for babies.