# Higher order inflection points

Consider a smooth plane curve $$X\subset\mathbb{P}^2$$ of degree $$d$$. We will say that $$x\in X$$ is an inflection point of order $$s$$ if the tangent line $$T_xX$$, of $$X$$ at $$x\in X$$, intersects $$X$$ in $$x\in X$$ with multiplicity at least $$s$$.

For instance, any point of $$X$$ in an inflection point of order $$s = 2$$, and the inflection points of order $$s = 3$$ are the flexes of $$X$$.

Does there exist a closed formula for the number $$I(s)$$ of inflection points of order $$s$$ of $$X$$. For instance, it is well known that $$I(3) = 3d(d-2)$$ since flexes are the intersection points of $$C$$ with its Hessian which is a curve of degree $$3(d-2)$$.

Furthemore, we must have $$I(s) = 0$$ for $$s > d$$ since otherwise $$C$$ would be reducible.

• As pointed out already, the general curve will not have a hyperflex (ie an inflection point of order $\geq4$). The loci in the parameter space $|\mathcal{O}_{\mathbb{P}^2}(d)|$ which do are studied by considering the bundle of principal parts and computing its Chern classes. This is explained really well in the "3264" book by Eisenbud-Harris – Frank Feb 4 at 8:19

(The statements you quote are only true if you are working over a field of characteristic zero or $$p > d$$. I will continue to make that assumption)
The formula is not $$I(3)=3d(d-2)$$ but rather $$\sum_{s>2} (s-2)I(s) = 3d(d-2)$$ (changing notation slightly so $$I(s)$$ counts the points with contact exactly $$s$$ instead of at least $$s$$).
The typical curve (non-empty open set of the parameter space) does indeed satisfy $$I(3)=3d(d-2)$$ and $$I(s)=0, s > 3$$. I think you can expect many different possibilities under the above constraints but most likely not all of them. I don't this has been worked out in general. For $$d=4$$ all possibilities and the dimension of the space of curves that satisfy them has been worked out in the PhD thesis of Vermeulen. I have unfortunately misplaced my copy so I can't quote the exact result.
Edit: Found the thesis (Amsterdam 1983). If $$d=4$$ then $$I(3)+2I(4)=24$$ and $$I(4)$$ can take any value between $$0$$ and $$12$$, except for $$10$$ and $$11$$.