Number of double tangents to an algebraic curve of degree d

Question

Let $C$ be a plane real algebraic curve of degree $d$, i.e., the zero-set of a two-variable polynomial of degree $d$.

Q1. Is it the case that the number of double tangents is $O(d^2)$?

I believe this follows from the Plücker formula. In fact, $d (d-1)$ is a tight upper bound. Am I reading this correctly? (My knowledge of algebraic geometry is thin.)

Q2. My actual situation is that my curve $C$ is (a) connected, and (b) embedded, that is, it is non-self-intersecting. Is there a smaller upper bound on the number of double tangents with these constraints?

If not, I would be interested in an example that achieves $\Omega(d^2)$ double tangents.

Answer 1

The number of bitangents to a plane curve (from the documentation of the maple package schubert).

proj(2,h,all):             # the dual projective plane
proj(2,j,all):             # the projective plane
bundlesection(C,o(d*j)):   # define a plane curve of degree d
morphism(f,C,Ph,[(d-1)*j]):# the Gauss map to the dual plane
multiplepoint(f,2)/2:      # double points are bitangents or flexes
                           # correct for flexes.  Flexes can of 
                           # course be calculated automatically.
bitangents:=expand(integral(C,%)-(3*d*(d-2)));
                                   4    3        2
                bitangents := 1/2 d  - d  - 9/2 d  + 9 d

subs(d=4,%);
                                   28