The existence of plane curves with prescribed singularities
Let \(K\) be field of characteristic \(0\). Let \(d, m_1,\ldots,m_r\) \((1 \leq r)\) be non-negative integers and \(P_1,\ldots,P_r \in \mathbb P_K^2\). The following problem is studied. When does there exist a reduced irreducible plane curve of degree \(d\) with an ordinary singular point of multiplicity \(m_i\) at \(P_i\) for all \(i\) and no other singularities? Let \(L\) be a line in \(\mathbb P^2\), \(L_{ij}\) the line through \(P_i\) and \(P_j\), \(I(L) = \{i |\, P_i \in L \}\) and assume \(\{ P_i |\, i \not\in I(L) \}\) are in general position. Let \(M(d) = [\frac {d^2} {3}] + 2d +1\). It is proved that such a curve exists being transversally to \(L\) and \(L_{ij}\) for all \(i,j\) if the following conditions are fulfilled: [(a)] \(\sum m_i(m_i+1)/2 \leq M(d-1)\), [(b)] \(\sum_{j \in J}m_j \leq d\) for all \(J \subset \{1,\ldots,r\}, | J| =2\), [(c)] \(\sum_{j \in I(L)}m_j \leq d\), [(d)] \(\sum_{j \in J}m_j \leq 2d-1\) for all \(I(L) \subset J\) with \(| J| =| I(L)| +2\), [(e)] \(r>1\) or \(m_1 \neq d\).
scientific article; zbMATH DE number 2058319