A bound for the Milnor sum of projective plane curves in terms of GIT (Q2813363)

The maximal total Milnor number \((d-1)^2\) of a plane curve of degree \(d \) is realised by concurrent lines. The next highest value occurs for Płoski curves: in even degree \(d/2\) conics intersecting in only one point, in odd degree such conics together with the tangent line in the intersection point. Using the degree of the polar map, the author proves that for stable curves (in the sense of GIT) of degree at least 5 the total Milnor number is at most \((d-1)^2-(d-2)\). If all irreducible components have degree at least 3, then the bound is \((d-1)^2-\lceil \frac{2d}3 \rceil\).