Higher order degrees of affine plane curve complements

Vanishing of higher order Alexander‐type invariants of plane curves

“…Remark We are using the convention $$\deg 0=\infty$$. The proof of [6, Theorem 5.18] assumes $$\delta _0(C)$$ is finite, but the result is also true for $$\delta _0(C)=\infty$$ because $$$$\begin{equation*} \delta _0(C)=\infty \Leftrightarrow F_1{\left(H_1(U,u_0;R_0)\right)}=0 \Leftrightarrow F_1{\left(H_1(U,u_0;\mathbb {Z}\Gamma _0)\right)}=0\Leftrightarrow \Delta _C^{\operatorname{multi}}=0. \end{equation*}$$$$In this list of equivalences, we have used that the projection $$G\twoheadrightarrow \Gamma _0$$ is the abelianization morphism and that R 0 is flat as a $$\mathbb {Z}\Gamma _0$$‐module.…”

“…Let $$\mathcal {V}_1(U)$$ be the first homology jump loci of U , namely, $$$$\begin{equation*} \mathcal {V}_1(U)=\lbrace \rho \in \mathrm{Hom}(G,\mathbb {C}^*)\mid H_1(U,\mathbb {C}_\rho )\ne 0\rbrace , \end{equation*}$$$$where $$\mathbb {C}_\rho$$ is the rank one $$\mathbb {C}$$‐local system on U induced by ρ. By [6, Remark 5.14], one has the following: $$$$\begin{equation*} \mathcal {V}_1(U)=(\mathbb {C}^*)^s\Leftrightarrow \text{ all the codimension }1\text{ minors of }B(0)\text{ are 0}. \end{equation*}$$$$This last condition is equivalent to the rank of the left $$\mathcal {K}_0$$‐module generated by the rows of B (0) being strictly smaller than $$m-1$$, which by Remark 2.3 is equivalent to $$\delta _0(C)=\infty$$.…”

“…However, if C is irreducible, both definitions coincide. Indeed, from [6, Remark 5.10], we know that $$\Delta ^{\operatorname{multi}}_C(t)$$ divides $$\Delta ^{\operatorname{uni}}_C(t)$$ in $$\mathbb {Q}[t^{\pm 1}]$$, and the same argument of the proof of [6, Theorem 5.18] (for $$s=1$$) shows that both polynomials are the same up to multiplication by a unit in $$\mathbb {Q}[t^{\pm 1}]$$.…”

“…These integers can also be interpreted as 𝐿 2 -Betti numbers associated to the tower of coverings of ℂ 2 ⧵ 𝐶 corresponding to the subgroups 𝐺 (𝑖) 𝑟 (the first of which is the universal abelian cover), so in principle, there is no reason to expect that such invariants have any good vanishing or finiteness properties. Some finiteness results obtained in [6,10] are summarized in this theorem.…”

“…This provides a broad generalization of the vanishing results of [6], where the fundamental group of one of the curve complements was assumed to be isomorphic to $$\mathbb {Z}$$, and $$\delta _n$$ of the other curve was assumed to be finite.…”