Mario Borelli

Emeritus - Associate

Education

B.A., Scuola Normale Superior, 1956
Ph.D., Indiana University, 1961

Research Group

Algebra and Algebraic Geometry

Bio

Research Interests

My research in Algebraic Geometry dealt with intersection theory and the characterization of projective varieties. In 1963 I defined the notion of Divisorial Varieties, and, in 1965, S. Kleiman proved that for divisorial varieties the Chevalley conjecture holds. I also was able to prove the amusing result that, again for divisorialvarieties, the statement “A Cartier divisor has an affine complement” (in the language of classical Italian geometry) is a linear condition. Part of Kleiman’s work is the proof of the statement:

“An algebraic variety is projective iff the codimension C of the cone of numerically non-negative Cartier divisors in the vector space of numerically equivalent Cartier divisors is 0.” Since there exist several examples of non-projective varieties, a question worth pursuing is the behavior of C with respect to various morphisms, products and immersions.

My interest in Computer Graphics led me to the development of a program package which graphs any two-, or three-, or four dimensional parametric representations of curves and surfaces, viewed from arbitrary user-selected viewpoints,using parallel or perspective projections with rather efficient hidden-line and contour-line algorithms. Both linear and bi-cubic approximation of surfaces can be used in the package.

Selected Publications

• Borelli, M.Variedades casi-proyectivas y divisoriales. Publicationes del Institutlo de Matematicas de la Universidad Nacional Interamericana, Lima, Peru, in press.
• Borelli, M. The cohomology of divisorial schemes. Pacific J. Math. 37:1–7, l97l.
• Borelli, M. Affine complements of divisors. Pacific J. Math. 3l : 595–605, 1969.
• Divisorial Varieties, Pacific. J. Math., 13 (1963), 375-388.
• Some results on ampleness and divisorial schemes, Pacific J. Math., 23 (1967), 217-227
• Topics in Local Algebra, by Jean Dieudonné, Notes by M. Borelli, Notre Dame Mathematical Lectures Number 10 (1967).