Construction of Calabi-yau 3-folds in P 6

We announce here the construction of examples of smooth Calabi-Yau 3-folds in P6 of low degree, up to degree 17. In the last degree their construction is rather complicated, and parametrized by smooth septics in P2 having a a g1 d with d = 13, 12, or 10. This turns out to show the existence of three unirational components of their Hilbert scheme, all having the same dimension 23+ 48 = 71. The constructions are based on the Pfaffian complex, choosing an appropriate vector bundle starting from their cohomology table. This translates into studying the possible structures of their Hartshorne-Rao modules. We also give a criterium to check the smoothness of 3-folds in P6. Constructions of smooth subvarieties of codimension 2 via a computer-algebra program have been extensively studied in recent years, mainly following the ideas presented in [4]. There the authors explicitely provide many constructions of surfaces in P4, showing that the problem to fill out all possible surfaces in P4 not of general type was indeed affordable, and this brought to a wide series of papers with similar examples. The starting point of these construction is based on the fact that a globalized form of the Hilbert-Burch theorem allows one to realize any codimension 2 locally Cohen-Macaulay subscheme as the degeneracy locus of a map of vector bundles. Precisely, for every codimension 2 subvariety X in Pn there is a short exact sequence 0 → F φ → G ψ → OPn → OX → 0, where F and G are vector bundles with rkG = rkF + 1 and ψ is locally given by the maximal minors of φ taken with alternating signs. In codimension 3 the situation is more complicated. Indeed in the local setting the minimal free resolution of every Gorenstein codimension 3 quotient ring of a regular local ring is given by a Pfaffian complex [1], but by globalizing this construction one obtains only the so called Pfaffian subschemes, i.e. subschemes defined locally by the 2r × 2r Pfaffians of an alternating map φ from a vector bundle of odd rank 2r + 1 to a twist of its dual. In particular, a Pfaffian subscheme in Pn has the following resolution: 0 → OPn (−t − 2s) ψ t → E(−t − s) φ → E(−s) ψ → OPn → OX → 0, where the map ψ is locally given by the 2r × 2r Pfaffians of φ and ψ t is the transposed of ψ . Being Pfaffian, this subscheme is automatically subcanonical, in the sense that its canonical bundle is the restriction of a multiple of OPn (1). A recent result of Walter [11] shows that under a mild additional hypothesis every subcanonical Gorenstein codimension 3 subscheme X in Pn is Pfaffian (see [5] for a description of the non-Pfaffian case), and therefore one can attempt to get its equations starting from constructing its Pfaffian resolution. †Short abstract version of the paper [10]