A flexible affine 𝑀-sextic which is algebraically unrealizable

Abstract

We prove that the union of a real algebraic curve of degree six and a real line on RP 2 cannot be isotopic to the arrangement in Figure 1. Previously, the second author realized this arrangement with flexible curves. Here we show that these flexible curves are pseudo-holomorphic in a suitable tame almost complex structure on CP 2 .For the proof of the algebraic non-realizability we consider all possible positions of the curve with respect to certain pencils of lines. Using the Murasugi-Tristram inequality for certain links in S 3 , we show that all the positions but one are unrealizable. Then, we prohibit the last position (the one which is realizable by a flexible curve) by studying its behaviour with respect to an auxiliary pencil of cubics.

Arrangements of an $M$-quintic with respect to a conic that maximally intersects its odd branch

“…The only exception is the arrangement denoted in [9] by D4. Its unrealizability was derived in [9] from the algebraic unrealizability of affine sextics of the isotopy type B 2 (1, 4, 5), proved in [4]. This argument cannot work in the pseudoholomorphic case for the simple reason that such an affine pseudoholomorphic sextic was constructed in [13] (see also [4]).…”

“…Its unrealizability was derived in [9] from the algebraic unrealizability of affine sextics of the isotopy type B 2 (1, 4, 5), proved in [4]. This argument cannot work in the pseudoholomorphic case for the simple reason that such an affine pseudoholomorphic sextic was constructed in [13] (see also [4]). All proofs in [9] become valid in the pseudoholomorphic case if the following line is added to Table 2 in the paper [9]:…”

“…Lemma 2.11. Let C, G, and F be real curves of bidegrees (2,8), (1,4), and (0, 1), respectively, on F 4 (G is a section and F is a fiber of . They can also be produced by applying a general method of construction of trigonal curves on ruled surfaces [17].…”

“…We blow up this point, and after that, blow down the proper transform of the fiber passing through it. This transforms C, G, and F into curves C , G , and F on F 3 of bidegrees (2, 7), (1,4), and (0, 1), respectively. Their arrangement on F 3 is depicted in Figure 10.1 for the choice of the leftmost oval in Figure 8.1.…”

“…Blowing up p, we obtain Figure 22.4, where the exceptional divisor is denoted by P . Blowing up p and then blowing down L 1 , we obtain a curve C of bidegree (4,8) on F 2 (see Definition 2.10), depicted in Figure 22.5. This curve has a singular point of type D 4 (a simple triple point) on the line P .…”