Conditions imposed by tacnodes and cusps (Q2750928)

Fix \(t\) points \(P_1,\dots,P_t\) of the complex projective plane \(\mathbb{P}^2\). For each \(P_i\) take either a zero-dimensional subscheme \(Z_i\) of \(\mathbb{P}^2\) with \((Z_i)_{\text{red}}=P_i\) or a cluster of a plane singularity. Set \(z_i:=\deg(Z_i)\), \(Z=Z_1\cup \cdots\cup Z_t\) and \(z=z_1+ \cdots+z_t= \deg (Z)\). For any integer \(d\) let \(\rho(Z,d): H^0(\mathbb{P}^2,{\mathcal O}_{\mathbb{P}^2} (d))\to H^0(Z,{\mathcal O}_Z(d))\) be the restriction map. One expects that if the points \(P_1, \dots,P_t\) are general, then the map \(\rho(Z,d)\) has maximal rank, i.e. \(\dim (\text{ker}(\rho(Z,d))) =\max\{0,(d+2) (d+1)/2-z\}\), except in a few controlled cases. Conjectures by Hirschowitz, Harbourne, Alexander, Ciliberto, Miranda, Greuel, Lossen, Shustin and others make explicit this expectation and several papers are devoted to prove it in some cases. In the present paper the author considers clusters corresponding to higher order cusps and tacnodes \((A_k\)-singularities) and to \(D_k\)-singularities. The positive results obtained here allow him to prove the existence of plane curves of low degree with tacnodes and higher order cusps.