On plane curves with double and triple points (Q2818766)

Let \(f\in S=\mathbb C[x,y,z]\) be homogeneous of degree \(N\), let \(C\subset\mathbb P^2\) be the projective curve defined by \(f\), and set \(U:=\mathbb P\setminus C\), a smooth affine variety. By \textit{P. Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 40, 5--57 (1971; Zbl 0219.14007)] \(H^\ast(U,\mathbb C)\) has a mixed Hodge structure, given by the Hodge filtration \(F\) and the polar filtration \(P\), and they agree on \(H^2(U,\mathbb C)\) for nodal curves. This result does not hold for non-nodal curves [see below]. \textit{A. Dimca} and \textit{G. Sticlaru} [Math. Proc. Camb. Philos. Soc. 153, No. 3, 385--397 (2012; Zbl 1253.14032)] have studied the case where \(C\) has only nodes as singularities. The first main result of this paper is Theorem 2., which treats the case of curves having nodes and triple points as singularities. There are formulae for the dimension of \(\mathrm{Gr}^1_F H^2(U,\mathbb C)\) and \(\mathrm{Gr}^2_F H^2(U,\mathbb C)\). Let \(M(f)=S/J_f\) be the Milnor algebra. In Theorem. 4.1, the main result of this paper, the authors give an estimation of the dimensions of certain homogeneous components of \(M(f)\). As a consequence, they find examples of curves with only ordinary double and triple points for which \(P^2H^2(U,\mathbb C)\neq F^2H^2(U,\mathbb C)\).