Resolution of Singularities of Surfaces by P. Del Pezzo. A Mathematical Controversy with C. Segre

We examine in some detail a series of papers published by P. Del Pezzo and Segre on the resolution of singularities of algebraic surfaces, trying to point out the main ideas. In 1888 P. Del Pezzo published a proof of the resolution of singularities of algebraic surfaces using birational transformations. He introduced the notion of "equisingularity" of two surfaces at a point or along a curve. This notion is made specific in a paper of 1889. C. Segre in his famous paper on "infinitely near multiple points" on a surface published in 1896 shows that Del Pezzo's method does not always produce a smooth surface. During 1897 each author reasserts his point of view in several papers which contain interesting remarks, regardless of their increasingly polemic style.