Strict transforms of plane curves record tangent directions

Write $f=f_m+f_{m+1}+\cdots$ as in [F1]. By [F2] the two charts cover $S^{\mathrm{\prime }}$ and meet in the locus $st=1$, and by [F3] the strict transform is cut out in them by the equations $g=0$ and $h=0$, where $g(0,s)=f_m(1,s)$ and $h(t,0)=f_m(t,1)$; thus $C^{\mathrm{\prime }}\cap E$ is computed in the first chart by the pair of equations $x=0$, $g=0$ and in the second by $y=0$, $h=0$.

In the first chart $C^{\mathrm{\prime }}\cap E$ is $\mathrm{Spec}k[s]\mathrm{/}(g(0,s))=\mathrm{Spec}k[s]\mathrm{/}(f_m(1,s))$, and in the second chart it is $\mathrm{Spec}k[t]\mathrm{/}(f_m(t,1))$. These are exactly the standard charts $D_{+}(u)$ and $D_{+}(v)$ of the closed subscheme $Z=\mathrm{Proj}(k[U,V]\mathrm{/}(f_m))\subseteq {\mathbb{P}}_k^1=\mathrm{Proj}k[U,V]$: on $D_{+}(u)$ one has $s=v\mathrm{/}u$ and the defining equation $f_m(1,s)=0$, and on $D_{+}(v)$ one has $t=u\mathrm{/}v$ and $f_m(t,1)=0$, with the overlap inverting $s$ and $t$. Hence $C^{\mathrm{\prime }}\cap E\cong Z$ as closed subschemes of $E$, the closed subscheme cut out by the form $f_m(u,v)$.

By [F5] the ring $k[U,V]\mathrm{/}(f_m)$ is the tangent cone ring of $C$ at the origin, so $Z=\mathrm{Proj}({\mathrm{Cone}}_0(C))$ is the projectivized tangent cone. Its closed points are the homogeneous prime ideals of $k[U,V]$ containing $f_m$ and not the irrelevant ideal, that is, the irreducible factors of $f_m$; writing $f_m={\prod }_i{p}_i^{s_i}$ with $p_i$ irreducible homogeneous of degree $d_i$, the point $q_i$ defined by $p_i$ has residue field of degree $d_i$ over $k$. For a point $q_i$ lying in the first chart, that is $p_i\mathrm{\ne }U$, the local ring of $Z$ at $q_i$ is $k[s{]}_{(p_i(1,s))}\mathrm{/}(f_m(1,s))$, whose length as an ${\mathcal{O}}_{Z,q_i}$-module is the exponent $s_i$; the point at infinity is computed in the second chart with the roles of $U$ and $V$ exchanged. So a factor of multiplicity $s$ contributes to $C^{\mathrm{\prime }}\cap E$ with multiplicity $s$. Over a splitting field of $f_m$ the factors $p_i$ are linear forms and the points $q_i$ are exactly the tangent directions of $C$ at the origin, with these multiplicities.

The underlying $0$-cycle of $C^{\mathrm{\prime }}\cap E$ has total degree $m$ over $k$: by [F4] the intersection is the $0$-cycle of degree $m$ cut out by the leading form, the origin being $k$-rational. Equivalently, the degrees of the points $q_i$ weighted by the multiplicities $s_i$ add up to $m$, matching the computation in the two charts of step 1.2.

Transversality in the squarefree case. Suppose $f_m$ is squarefree, so its irreducible factors occur with multiplicity one; this covers a node and a cusp with reduced tangent cone, where the leading form is a product of distinct linear or irreducible factors. Let $q$ be a closed point of $C^{\mathrm{\prime }}\cap E$ lying in the first chart and let $p(s)$ be the corresponding irreducible factor of $f_m(1,s)$, which is simple; the case of a point lying only in the second chart is symmetric. Write $g(x,s)=p(s)u(s)+xw(x,s)$ with $u(s)$ a unit at $p$, which is possible because $g(0,s)=f_m(1,s)=p(s)u(s)$ and $p$ is simple. In the local ring ${\mathcal{O}}_{S^{\mathrm{\prime }},q}$ with maximal ideal $\mathfrak{m}=(x,p(s))$, the equation $g$ lies in $\mathfrak{m}\setminus {\mathfrak{m}}^2$ and the quotient ${\mathcal{O}}_{C^{\mathrm{\prime }},q}={\mathcal{O}}_{S^{\mathrm{\prime }},q}\mathrm{/}(g)$ has maximal ideal generated by $x$, because $p(s)u(s)\equiv -xw(x,s)$ modulo $g$ and $u$ is a unit; hence ${\mathcal{O}}_{C^{\mathrm{\prime }},q}$ is a regular one-dimensional local ring and $n_q(C^{\mathrm{\prime }},E)={\mathrm{length}}_{{\mathcal{O}}_{C^{\mathrm{\prime }},q}}\left({\mathcal{O}}_{C^{\mathrm{\prime }},q}\mathrm{/}x{\mathcal{O}}_{C^{\mathrm{\prime }},q}\right)=1$. By [F8] contact order one is exactly transversality at $q$, so $C^{\mathrm{\prime }}$ and $E$ meet transversally at every point of $C^{\mathrm{\prime }}\cap E$ when $f_m$ is squarefree.

Steps 1.2, 2.1, 3.1 and 3.2 prove all the assertions: $C^{\mathrm{\prime }}\cap E$ is the closed subscheme of $E={\mathbb{P}}_k^1$ cut out by the form $f_m(u,v)$, its closed points are the irreducible factors of $f_m$ with the corresponding multiplicities, the underlying $0$-cycle has total degree $m$ over $k$, the points are the tangent directions of $C$ at the origin with multiplicity over a splitting field, and for squarefree $f_m$ the intersection is transverse at every point.