Symmetric plane curves of degree 7: pseudoholomorphic and algebraic classifications

Abstract

This paper is motivated by the real symplectic isotopy problem : does there exist a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real schemes (resp. complex schemes) realizable by symmetric real curves of degree 7 with respect to the type of the curve (resp. M -symmetric real curves of degree 7). In particular, we exhibit two real schemes which are realizable by real symmetric dividing pseudoholomorphic curves of degree 7 on the projective plane but not by algebraic ones.

A nonalgebraic patchwork

“…A lot of examples are known of nonsingular real pseudoholomorphic curves in R n which are L-isotopic to no homologous real algebraic curves (see for example [4,6,17,18]). However, as far as we know none of those examples are constructed with the pseudoholomorphic patchworking of Itenberg and Shustin, and the question of the existence of a patchworked pseudoholomorphic curves with any kind of nonalgebraic behavior was open.…”

Corrigendum to “A flexible affine 𝑀-sextic which is algebraically unrealizable”

“…The main theorem of the paper [4] states that the arrangement B 2 (1,4,5) in RP 2 (see Figure 1) is unrealizable by a union of a line and a real smooth algebraic sextic curve. The precise statement is: Theorem 1.…”

“…Then there does not exist an ambient isotopy of RP 2 which deforms C and L into the curve and the line in Figure 1. (1,4,5) Recently, the first author found a mistake in the final part of the proof of Theorem 1 given in [4] (we discuss this mistake in detail in Section 3 below). However the result is correct and here we give another proof.…”

“…Typeset by A M S-T E X Three arrangements are realizable pseudo-holomorphically, but not algebraically; see [12]. The arrangement B 2 (1,4,5) in Figure 1 is one of them. This is why it is more difficult to exclude it.…”

“…(i) assuming that a smooth sextic curve C 0 arranged with respect to the line L as shown in Figure 1 exists, we derive that there exists a real elliptic sextic curve C 9 with 9 nodes located with respect to L as shown in Figure 2(a) (see Lemma 1 in Section 1); (ii) from the existence of a sextic C 9 we derive the existence of an elliptic real sextic having 7 nodes (five isolated and two non-isolated) and a singularity A 3 , and located with respect to L as shown in Figure 2(b)(see Lemma 2 in Section 1); (iii) we prohibit the existence of the latter real elliptic sextic using a suitable version of cubic resolvent (see Section 2). So, the general scheme of the proof of Theorem 1 is almost the same as for the proof in [12] of algebraic unrealizability of the affine sextic C 2 (1,3,6). However, there is a difference in the last step.…”