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!
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1$\begingroup$ Which article are you reading, precisely? $\endgroup$– Francesco PolizziAug 4, 2011 at 7:58
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$\begingroup$ Are you sure it said that? It's not proper English. $\endgroup$– TonyKAug 4, 2011 at 11:52
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$\begingroup$ Why is this being downvoted? Google finds several books containing the phrase "consecutive double/triple points". $\endgroup$– user2035Aug 4, 2011 at 14:26
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2$\begingroup$ Just a guess: maybe a "consecutive triple point" is a (3,3) point, namely a triple point with another triple point infinitely near to it, as $(0,0)\in\{x3+y^6=0\}$. $\endgroup$– ritaAug 4, 2011 at 16:05
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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.
