부산대학교 도서관

For a surface $X\subset{\Bbb P}^N$, this article considers two types of generic projections: \roster \item A generic projection ${\Bbb P}^N\dashrightarrow {\Bbb P}^3$ induces a birational morphism $X\to X'\subset{\Bbb P}^3$. It is well known that the singularities of $X'$ consist of a double curve $E$, as well as double points and triple points that lie on $E$. \item A generic projection ${\Bbb P}^N\dashrightarrow {\Bbb P}^2$ induces a finite morphism $\pi\:X\to{\Bbb P}^2$. It is well known that the branch curve $B$ of $\pi$ is irreducible with simple nodes and cusps as singularities. \endroster \par A classical theorem of B. Segre [Mem. Accad. Italia {\bf 1} (1930), no. 4, 31 pp.; JFM 56.0562.01] characterizes precisely which curves arise as branch curves of generic projections from surfaces in ${\Bbb P}^3$. This result was re-proven in Part I [M. Friedman, M. Leyenson and E.~I. Shustin, Internat. J. Math. {\bf 22} (2011), no.~5, 619--653; MR2799882 (2012h:14084)]. \par In the article under review, the authors generalize this result to generic projections from surfaces $X\subset{\Bbb P}^N$ with arbitrary $N\geq3$. More precisely, they consider a composite of generic projections ${\Bbb P}^N\dashrightarrow {\Bbb P}^3\dashrightarrow{\Bbb P}^2$. Then, they form the union $C\coloneq B\cup F\subset{\Bbb P}^2$, where $B$ is the branch curve of the induced generic projection $\pi\:X\to{\Bbb P}^2$ and $F$ is the image of the double curve $E\subset X'\subset{\Bbb P}^3$ arising from the induced generic projection $X\to X'\subset{\Bbb P}^3$. Their main result is Theorem 4.1, which gives numerical conditions on a union of two plane curves (similar to the classical ones of Segre, but much more involved) that are necessary and sufficient for it to arise from such a composite generic projection.