Gallarati undertook research at the l'Instituto Nazionale di Alta Matematica (The National Institute of Higher Mathematics) in Rome. There he was taught by most of the stars of Italian mathematics of this period: Giacomo Albanese, Leonard Roth (teaching in Rome as a visiting professor from Imperial College, London), L Tonelli, E G Togliatti, Beniamino Segre, and Francesco Severi.
Gallarati was appointed to the University of Genova in 1947 and he remained there until he retired in 1987.
In , 33 of the 64 papers which Gallarati published between 1951 and 1996 are reproduced. These reflect the major area of his research which was mostly in algebraic geometry. Eight of the papers reproduced in  study surfaces in P3 with many isolated singularities. This was one of Gallarati's most important mathematical contributions. It is worth noting that some of lower bounds obtained by Gallarati for the maximal number of nodes of surfaces of degree n were not improved until comparatively recently and for large values of n the exact maximal number of nodes of surfaces of degree n is still unknown despite much recent work on this topic.
Three papers given in  relate to Gallarati's contributions to Grassmannian geometry. His work on this topic extended in several ways the bound obtained by Beniamino Segre for the number of linearly independent linear complexes containing the curve in the Grassmannian corresponding to the tangent lines of a nondegenerate projective curve. Gallarati extended the results to the case of tangent spaces of varieties of arbitrary dimensions, to arbitrary curves in Grassmannians corresponding to nondegenerate scrolls, and to complexes of higher degree.
Gallarati studied properties of varieties whose tangent spaces meet certain linear subspaces along spaces of dimension higher than expected. He gave an elegant characterization of the Veronese and Segre varieties in terms of their tangential properties. Another particularly notable research contribution was made by Gallarati in his work on multiple contacts of surfaces along a curve. He constructed counterexamples to conjectures of Babbage and Hochster and gave negative answers to questions posed by Mumford and Hartshorne.
Other work by Gallarati included a classification of Fano varieties of the second kind, a study of irregularity of double spaces, and the finding of bounds for the class of a surface with ordinary singularities in terms of its degree.
Zak, reviewing , remarks that republishing Gallarati's papers:-
... introduces the modern reader to the almost forgotten world of projective algebraic geometry, Italian style.Gallarati taught algebraic geometry for many years and notes from these courses still exist :-
These notes reflect the unique style of their author and are rich in original observations.
Article by: J J O'Connor and E F Robertson