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We generalize the classical approach of describing the infinitesimal Torellimap in terms of multiplication in a Jacobi ring to the case of quasi-smoothcomplete intersections in weighted projective space. As an application, weprove the infinitesimal Torelli theorem for hyperelliptic Fano threefolds ofPicard rank 1, index 1, degree 4 and study the action of the automorphism groupon cohomology. The results of this paper are used to prove Lang-Vojta'sconjecture for the moduli of such Fano threefolds in a follow-up paper.

Infinitesimal Torelli for weighted complete intersections and certain Fano threefolds