# 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).

**Email:** mario@nd.edu

**Phone:** 574-631-7334

**Fax:** 574-631-6579

**Office:** 161 Hurley Hall