Generalized adjoint forms on algebraic varieties

TL;DR

This paper generalizes classical results on algebraic varieties, extending the Castelnuovo's free pencil trick and exploring analogies with the Adjoint Theorem, while also providing a new formulation of Griffiths's infinitesimal Torelli Theorem for hypersurfaces.

Contribution

It introduces a comprehensive generalization of the Castelnuovo's free pencil trick and offers a novel formulation of Griffiths's infinitesimal Torelli Theorem for smooth projective hypersurfaces.

Findings

Generalized Castelnuovo's free pencil trick

Analogies with the Adjoint Theorem established

New formulation of Griffiths's infinitesimal Torelli Theorem

Abstract

We prove a full generalization of the Castelnuovo's free pencil trick. We show its analogies with the Adjoint Theorem; see L. Rizzi, F. Zucconi, Differential forms and quadrics of the canonical image, arXiv:1409.1826 and also Theorem 1.5.1 in G. P. Pirola, F. Zucconi, Variations of the Albanese morphisms, J. Algebraic Geom. 12 (2003), no. 3, 535-572. Moreover we find a new formulation of the Griffiths's infinitesimal Torelli Theorem for smooth projective hypersurfaces using meromorphic $1$-forms.