Tangent Hyperplane $H$ to $X$ at $p \in X$ and hyperplane divisor $\operatorname{div}(H)$.

Question

According to Rick Miranda (Algebraic Curves and Riemann Surfaces) we have the following Lemmas:

Lemma 3.7 (page 219): Suppose that $X \subset \mathbb{P}^n$ is a nondegenerate smooth curve (with $n \ge 2$). Then $X$ has only finitely many flex points.

Lemma 3.8 (page 220): Suppose that $X \subset \mathbb{P}^n$ is a nondegenerate smooth curve (with $n \ge 2$). Then there are only finitely many pairs of distinct points $p$ and $q$ with the same tangent line.

The above lemmas combine to give the following corollary:

Corollary 3.9 (page 221): Suppose that $X \subset \mathbb{P}^n$ is a nondegenerate smooth curve of degree $d$ (with $n \ge 2$).

a) The general hyperplane $H$ in $\mathbb{P}^n$ is such that its divisor $\operatorname{div}(H)$ consists of $d$ distinct points $\{p_i\}$, each having

$\operatorname{div}(H)(p_i)=1$ (i.e., the general hyperplane is transverse to $X$).

b) For all but finitely many points $p$ of $X$, the general tangent hyperplane $H$ to $X$ at $p$ is such that

$\operatorname{div}(H) = 2\cdot p + q_3 + \cdots + q_d$ with all $q_i$ distinct and unequal to $p$ (i.e., $H$ is neither a flexed tangent nor a bitangent hyperplane).

My question is:

Affirmation: From these results it is correct to state that for a point $p$ in $X$ and a tangent hyperplane $H$ to $X$ in $p$ we have the following possibilities for hyperplane divisor $\operatorname{div}(H)$:

1. $\operatorname{div}(H)=2\cdot p + q_3 + \cdots + q_d$ with all $q_i$ distinct and unequal to $p$; in case $H$ is general tangent;

2. $\operatorname{div}(H)=k\cdot p + q_{k+1} + \cdots + q_d$ with all $q_i$ distinct and unequal to $p$; in case $p \in X$ is a flex point, i.e., $k \ge 3$;

3. $\operatorname{div}(H)=k_1\cdot p_1+ k_2\cdot p_2 + q_{k_1 + k_2 +1} + \cdots + q_d$ with all $q_i$ distinct and unequal to $p_1$ and $p_2$; in case $H$ is bitangent to $X$.

Is this Affirmation correct? The affirmation is it true due to Lemmas 3.7, Lemma 3.8 and Corollary 3.9?

Answer 1

No, this is not correct. Here is a counterexample: $$x^6+y^6+z^6=10(x^3y^3+y^3z^3+z^3x^3).$$

This is smooth, and it has 72 tritangents (lines tangent to the curve at 3 points).