what is the meaning of "consecutive triple points"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T15:31:56Z http://mathoverflow.net/feeds/question/72067 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72067/what-is-the-meaning-of-consecutive-triple-points what is the meaning of "consecutive triple points"? pmath 2011-08-04T06:21:13Z 2011-08-05T10:47:15Z <p>Hi, I am reading an the Shah's article " A complete moduli Space for K3 surfaces of degree 2" At some point, he analyses the singularities on plane curves of degree six. He uses the phrase: "Reduce sextics which have neither consecutive triple points.."[Th 2.4]. I am confuse for the sentence. What is the meaning of that? Are they fat points in some special configuration? Can anyone give me a polynomial that induces this kind of singularities? Thanks!</p> http://mathoverflow.net/questions/72067/what-is-the-meaning-of-consecutive-triple-points/72167#72167 Answer by rita for what is the meaning of "consecutive triple points"? rita 2011-08-05T10:47:15Z 2011-08-05T10:47:15Z <p>I've had a quick look at the paper and I can confirm that my guess in the comment above was correct.The sextics in Group I in the statement of Thm. 2.4 are those that have at most double points or triple points that do not have an infinitely near triple point. These singularities are discussed in II.8 of Barth-Peters-Van de Ven "Compact complex surfaces" (I have the old edition) where they are called simple singularities. Sometimes they are also called negligible singularities. The reason for the terminology is the following: if $C$ is a curve in a smooth surface $S$, $X\to S$ is a double cover branched on $C$ and $P\in C$ is a singular point, then $X$ has a rational double point over $P$ iff $P\in C$ is simple iff (locally near $P$) $K_Y= f^*K_X$ for the minimal resolution $f\colon Y\to X$. In old fashioned language, these singularities "do not impose adjunction conditions" on double covers. </p>