A note on splitting curves of plane quartics and multi-sections of rational elliptic surfaces

Two-graphs and the Embedded Topology of Smooth Quartics and its Bitangent Lines

“…Proof. The same statement for smooth cubic contact curves instead of L i +L j + L k is proved in [4,Proposition 3.3]. The same proof works in our case where L i + L j + L k is a reducible thus singular contact curve.…”

“…Proof. As in the proof of [4,Proposition 3.3], if there exists a conic C through the six points of tangency of Q and L i , L j , L k , the line L l through the remaining two intersection points of C and Q must be a bitangent line. The existence of the conic implies that the other three triples {L j , L k , L l }, {L i , L k , L l }, {L i , L j , L l } are also elements of ∆ I .…”

“…(We could have done the same for (6,14), (6,16), (6,20).) For the remaining cases (6, 0), (6,4), (6,6), (…”

“…Many invariants have been used to this end such as the fundamental groups of the complements π(P 2 \ C i ), the Alexander polynomials ∆ Ci (t) and the existence/non-existence of certain Galois covers branched along C i (See [2] for a survey on these topics). Also, other more recent types of invariants called "splitting invariants" have been developed and been proved effective in studying reducible curves ( [3], [4], [13], [14]). However, as the number of irreducible components increase, these invariants become more increasingly complex, and it becomes difficult to manage the data they provide.…”

Connected Numbers and the Embedded Topology of Plane Curves

“…In this paper, we study the embedded topologies of plane curves based on the ideas used in [5] and [17]. Here a plane curve C is a reduced divisor on the complex projective plane P 2 := P 2 C , and the embedded topology of the plane curve C is the homeomorphism class of the pair (P 2 , C).…”