# Mario Borelli

**Emeritus**

B.A., Scuola Normale Superior, 1956

Ph.D., Indiana University, 1961

Email: Mario.Borelli.1@nd.edu

Office: 116 Hayes-Healy Bldg

Phone:(574) 631-7334

Fax: (574) 631-6579

**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).

Please direct questions and comments to: Mario.Borelli.1@nd.edu