We give examples of smooth Calabi-Yau 3 3 -folds in P 6 P^6 of low degree, up to the first difficult case, which occurs in degree 17. In this case we show the existence of three unirational components of their Hilbert scheme, all having the same dimension 23 + 48 = 71 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 3 -folds in P 6 P^6 .
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Fabio Tonoli (2003) studied this question.
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