Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups

Abstract

We study the fundamental groups of (the complement of) six plane conic-line arrangements of degree 7. Those fundamental groups have a canonical quotient that is often a simple quotient of Coxeter groups. We consider the conic-line arrangements introduced by Tokunaga, which consist of a pair of conic-line arrangements with three conics in each (and thus, each has a single line) and a pair with two conics in each (and thus, each has three lines). Using the theory of Coxeter groups, we are able to show that the Zariski pairs of conic-line arrangements found by Tokunaga in 2014 have non-isomorphic fundamental groups. We also present one additional Zariski pair of conic-line arrangements found by our method. This pair has a single conic (and thus, each has five lines) in every arrangement. It also contains a complicated singularity that is neither a simple node nor a tangency, which requires a more careful local analysis.

Smooth Q$\mathbb {Q}$‐homology planes satisfying the negativity conjecture

“…We have shown that 𝑈 meets both (−2)-tips of 𝐷 am , in particular, 𝑈 ⊈ 𝐷 am . Now 𝑈 ⋅ 𝑅 am ⩽ 1 by formula (12), so in fact 𝑈 ⋅ 𝑅 am = 1 by condition (7), as claimed. □ Proposition 6.1 shows that, under our assumptions, 𝑋 min is ℙ(1, 2, 3), ℙ(1, 1, 2), or ℙ 2 .…”

“…Then 𝑈 ⋅ 𝑅 am ⩾ 1 because by Lemma 3.12(b) 𝑋 am ⧵ 𝐷 am contains no contractible curves. By the formula (12), 𝑈 ⋅ 𝑅 am = 1, hence 𝑈 is a bubble on (𝑋 am , 𝐷 am ), contrary to the condition (7). In case (b), formula (1) gives 𝑈 ⋅ Bk Δ am = 1, so 𝑈 ⋅ 𝑅 am = 0 by formula (12), and again 𝑈 is a bubble; a contradiction.…”

“…If 𝑈 ⊆ 𝑅 am then 𝛽 𝑅 am (𝑈) = 𝑈 ⋅ 𝑅 am − 𝑈 2 ⩽ 1 and 𝛽 𝐷 am (𝑈) = 𝛽 𝑅 am (𝑈) + 𝑈 ⋅ Δ am ⩽ 3, contrary to Lemma 3.12(e). Hence, 𝑈 ⊈ 𝑅 am , so 𝑈 ⋅ 𝑅 am = 0 and thus 𝑈 is a bubble on (𝑋 am , 𝐷 am ); a contradiction with condition (7).…”

“…In the latter case, Figure 2 shows some (−1)-curves on 𝑋 am . In Proposition 6.1, we will see that unless 𝑋 min ≅ ℙ(1, 2, 3), at least one of these curves is a bubble on (𝑋 am , 𝐷 am ), contrary to condition (7). Later, we will see that lack of bubbles restricts the geometry of 𝐷 am , too, see Example 4.5.…”

“…Looking at Figures 2b-2d again, we see that 𝑈 + ⋅ Δ am = 2, and 𝑈 + ⋅ Bk(Δ am − 𝑇) ⩾ 1 #𝑇+1 , so 𝑈 + ⋅ (Δ am − Bk Δ am ) ⩾ 1. By formula (12), 𝑈 + ⋅ 𝑅 am = 0, so 𝑈 + is a bubble; a contradiction with condition (7).…”