Fabio Tonoli
Construction of Calabi-Yau \\3-folds in ${\bf P}^6$
Preprint series: Mathematica Gottingensis
MSC:
14J10 Families, moduli, classification: algebraic theory
14J30 Special $3$-folds, See also {14E05}
Abstract: We give examples of smooth Calabi-Yau 3-folds in ${\bf P}^6$ 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 $23+48=71$.

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 ${\bf P}^6$.
Keywords: Calabi-Yau 3-folds, Pfaffian, unirationality, syzygies, finite fields