On the existence of maximizing curves of odd degrees
Marek Janasz, Izabela Leśniak
TL;DR
The paper addresses the existence of maximizing curves of odd degree and proves a non-existence criterion under ADE singularity constraints, using log canonical thresholds and Arnold exponents to bound the minimal Jacobian relation degree ${\rm mdr}(f)$; it further introduces $M$-curves as a broader, constructible class of free plane curves with extremal total Tjurina numbers and provides practical criteria and several explicit examples. By demonstrating that many candidate ADE configurations cannot realize maximizing curves in odd degrees, the work explains the rarity of such curves and offers constructive alternatives. It also extends the framework to even-degree curves through $M$-curves and explores M-line arrangements, providing combinatorial characterizations and verified instances via computational tools such as $\text{SINGULAR}$. Overall, the results connect singularity theory, freeness, and Tjurina invariants to illuminate both the constraints and opportunities for building extreme plane curves. The findings thus contribute both a rigorous non-existence criterion and a constructive path (via $M$-curves and M-line arrangements) for obtaining highly constrained, free plane curves with maximal or near-maximal total Tjurina numbers.
Abstract
In this paper we provide the non-existence criterion for the so-called maximizing curves of odd degrees. Furthermore, in the light of our criterion, we define a new class of plane curves that generalizes the notion of maximizing curves which we call as $M$-curves.