Past exam of the geometry course of the University of Oxford / 2008 / B / 6 / Solution

An inflection point is a smooth point where the tangent line has intersection multiplicity at least three with the curve. For a nonsingular plane curve of degree $n\ge 3$, these are the intersections with its Hessian curve, whose degree is $3(n-2)$. The original curve is not a component of its Hessian: otherwise in each graph chart the tangent direction would have zero derivative identically, making the curve locally a line and hence a line globally. Bézout therefore gives the upper bound $3n(n-2)$, counting multiplicity. A nonsingular conic has no inflection points; lines are excluded from this finite-contact definition.

Take $F=y^{2}z-x^3+x{z}^2$. Its Hessian determinant isAt infinity the curve has only $O=[0:1:0]$, which is an inflection point: its tangent is $z=0$ and the restricted equation is $-x^3=0$. In the chart $z=1$, substitute $y^2=x^3-x$ into the Hessian equation to obtainThus the full answer isThe four complex values of $x$ are distinct and none is $0,1,-1$, so each gives two distinct nonzero values of $y$. Together with $O$ these are all nine inflection points, exhausting the Bézout bound.