부산대학교 도서관

For a surface $X\subset{\Bbb P}^N$, this article considers two types ofgeneric projections:\roster\item Ageneric projection ${\Bbb P}^N\dashrightarrow {\Bbb P}^3$ induces abirational morphism $X\to X'\subset{\Bbb P}^3$. It is well known thatthe singularities of $X'$ consist of a double curve $E$, as well asdouble points and triple points that lie on $E$.\item Ageneric projection ${\Bbb P}^N\dashrightarrow {\Bbb P}^2$ induces afinite morphism $\pi\:X\to{\Bbb P}^2$. It is well known that the branchcurve $B$ of $\pi$ is irreducible with simple nodes and cusps assingularities.\endroster\parA classical theorem of B. Segre [Mem. Accad.Italia {\bf 1} (1930), no. 4, 31 pp.; JFM 56.0562.01] characterizes preciselywhich curves ariseas 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)].\parIn the article under review, the authors generalize this result togeneric projections from surfaces $X\subset{\Bbb P}^N$ with arbitrary$N\geq3$. More precisely, they consider a composite of genericprojections ${\Bbb P}^N\dashrightarrow {\Bbb P}^3\dashrightarrow{\BbbP}^2$. 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 $E\subsetX'\subset{\Bbb P}^3$ arising from the induced generic projection $X\toX'\subset{\Bbb P}^3$. Their main result is Theorem 4.1, which givesnumerical conditions on a union of two plane curves (similar to theclassical ones of Segre, but much more involved) that are necessary andsufficient for it to arise from such a composite generic projection.