Examples of Type IV unprojection
نویسنده
چکیده
I show that P(2, 3) has an embedding P(2, 3) ∼= Γ ⊂ P(4, 5, 6, 9) whose image Γ is contained in a quasismooth K3 hypersurface X24 ⊂ P(4, 5, 6, 9). The pair Γ ⊂ X24 unprojects to the codimension 4 K3 surface Y ⊂ P(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) ]. Numerator = 1− t − t − 2t − 2t − 2t − t + t + 2t + 3t + 4t + 3t + · · · (Altınok4(111) in the Magma K3 database). The local coordinates at the third centre P3 = 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 P3. 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.
منابع مشابه
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Type III unprojection plays a very important role in the birational geometry of Fano threefolds (cf. [CPR], [Ki], [BZ]). According to [Ki] p. 43, it was first introduced by A. Corti on his calculations of Fano threefolds of genus 6 and 7. It seems that at present a general definition of type III unprojection is still missing. After proving in Section 2 some general facts about residual ideals, ...
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