The object of this problem is to find a property of the local ring
\(\mathscr{O}_{P}(F)\) that determines whether or not \(P\) is an ordinary
multiple point on \(F\).
Let \(F\) be an irreducible plane curve, \(P=(0,0), m=m_{P}(F)>1\). Let
\(\mathfrak{m}=\mathfrak{m}_{P}(F)\). For \(G \in k[X, Y]\), denote its residue in
\(\Gamma(F)\) by \(g ;\) and for \(g \in \mathfrak{m}\), denote its residue in
\(\mathfrak{m} / \mathrm{m}^{2}\) by \(\bar{g}\). (a) Show that the map from
\\{forms of degree 1 in \(k[X, Y]\\}\) to \(\mathfrak{m} / \mathrm{m}^{2}\) taking
\(a X+\) \(b Y\) to \(\overline{a x+b y}\) is an isomorphism of vector spaces (see
Problem 3.13). (b) Suppose \(P\) is an ordinary multiple point, with tangents
\(L_{1}, \ldots, L_{m}\). Show that \(I\left(P, F \cap L_{i}\right)>m\) and
\(\bar{l}_{i} \neq \lambda \overline{l_{j}}\) for all \(i \neq j\), all \(\lambda
\in k\). (c) Suppose there are \(G_{1}, \ldots, G_{m} \in k[X, Y]\) such that
\(I\left(P, F \cap G_{i}\right)>m\) and \(\bar{g}_{i} \neq \lambda \bar{g}_{j}\)
for all \(i \neq j\), and all \(\lambda \in k\). Show that \(P\) is an ordinary
multiple point on \(F .\) (Hint:: Write \(G_{i}=L_{i}+\) higher terms.
\(\bar{l}_{i}=\bar{g}_{i} \neq 0\), and \(L_{i}\) is the tangent to \(G_{i}\), so
\(L_{i}\) is tangent to \(F\) by Property (5) of intersection numbers. Thus \(F\)
has \(m\) tangents at \(P .\) ) (d) Show that \(P\) is an ordinary multiple point on
\(F\) if and only if there are \(g_{1}, \ldots, g_{m} \in \mathfrak{m}\) such that
\(\bar{g}_{i} \neq \lambda \bar{g}_{j}\) for all \(i \neq j, \lambda \in k\), and
\(\operatorname{dim} \mathscr{O}_{P}(F) /\left(g_{i}\right)>m\)
