Plame Curres. Let \(X\) be a curve of degree \(d\) in \(\mathbf{P}^{2}\). For each
point \(P \in X\), let \(T_{P}(X)\) be the tangent line to \(X\) at \(P(\mathrm{I},
\mathrm{Ex} .7 .3) .\) Considering \(T_{P}(X)\) as a point of the dual projective
plane \(\left(\mathbf{P}^{2}\right)^{*},\) the \(\operatorname{map} P \rightarrow
T_{P}(X)\) gives a morphism of \(X\) to its dual curre \(X^{*}\) in
\(\left(\mathbf{P}^{2}\right)^{*}\) (I, Ex. 7.3). Note that even though \(X\) is
nonsingular, \(X^{*}\) in general will have singularities. We assume char \(k=0\)
below.
(a) \(\mathrm{Fix}\) a line \(L \subseteq \mathrm{P}^{2}\) which is not tangent to
\(X .\) Define a morphism \(\varphi: X \rightarrow L \mathrm{by}\)
\(\varphi(P)=T_{P}(X) \cap L,\) for each point \(P \in X .\) Show that \(\varphi\)
is ramified at \(P\) if and only if either \((1) P \in L,\) or (2)\(P\) is an
inflection point of \(X,\) which means that the intersection multiplicity (I,
Ex. 5.4) of \(T_{P}(X)\) with \(X\) at \(P\) is \(\geqslant 3 .\) Conclude that \(X\)
has only finitely many inflection points.
(b) A line of \(\mathbf{P}^{2}\) is a multiple tangent of \(X\) if it is tangent
to \(X\) at more than one point. It is a bitangent if it is tangent to \(X\) at
exactly two points. If \(L\) is a multiple tangent of \(X\), tangent to \(X\) at the
points \(P_{1}, \ldots, P_{r},\) and if none of the \(P_{i}\) is an inflection
point, show that the corresponding point of the dual curve \(X^{*}\) is an
ordinary \(r\) -fold point, which means a point of multiplicity \(r\) with
distinct tangent directions \((\mathrm{I}, \mathrm{Ex} .5 .3) .\) Conclude that
\(X\) has only finitely many multiple tangents.
(c) Let \(O \in \mathbf{P}^{2}\) be a point which is not on \(X\), nor on any
inflectional or multiple tangent of \(X\). Let \(L\) be a line not containing \(O
.\) Let \(\psi: X \rightarrow L\) be the morphism defined by projection from \(O
.\) Show that \(\psi\) is ramified at a point \(P \in X\) if and only if the line
\(O P\) is tangent to \(X\) at \(P,\) and in that case the ramification index is 2.
Use Hurwitz's theorem and (I, Ex. 7.2) to conclude that there are exactly
\(d(d-1)\) tangents of \(X\) passing through \(O .\) Hence the degree of the dual
curve (sometimes called the class of \(X\) ) is \(d(d-1)\)
(d) Show that for all but a finite number of points of \(X,\) a point \(O\) of \(X\)
lies on exactly \((d+1)(d-2)\) tangents of \(X,\) not counting the tangent at \(O\)
(e) Show that the degree of the morphism \(\varphi\) of (a) is \(d(d-1)\).
Conclude that if \(d \geqslant 2 .\) then \(X\) has \(3 d(d-2)\) inflection points,
properly counted. (If \(T_{P}(X)\) has intersection multiplicity \(r\) with \(X\) at
\(P,\) then \(P\) should be counted \(r-2\) times as an inflection point. If \(r=3\)
we call it an ordinary inflection point.) Show that an ordinary inflection
point of \(X\) corresponds to an ordinary cusp of the dual curve \(X^{*}\)
(f) Now let \(X\) be a plane curve of degree \(d \geqslant 2,\) and assume that
the dual curve \(X^{*}\) has only nodes and ordinary cusps as singularities
(which should be true for sufficiently general \(X\) ). Then show that \(X\) has
exactly \(\frac{1}{2} d(d-2)(d-3)(d+3)\) bitangents. [Hint: Show that \(X\) is the
normalization of \(X^{*}\). Then calculate \(p_{a}\left(X^{*}\right)\) two ways:
once as a plane curve of degree \(d(d-1),\) and once using (Ex. \(1.8 \text { ).
}]\)
(g) For example, a plane cubic curve has exactly 9 inflection points, all
ordinary. The line joining any two of them intersects the curve in a third
one.
(h) A plane quartic curve has exactly 28 bitangents. (This holds even if the
curve has a tangent with four-fold contact, in which case the dual curve
\(X^{*}\) has a tacnode.
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