Bitangents of real algebraic curves: signed count and constructions

Abstract

We study real bitangents of real algebraic plane curves from two perspectives. We first show that there exists a signed count of such bitangents that only depends on the real topological type of the curve. From this follows that a generic real algebraic curve of even degree $d$ has at least $\frac{d(d-2)}{2}$ real bitangents. Next we explain how to locate (real) bitangents of a (real) perturbation of a multiple (real) conic in $\mathbb{C}P^2$. As main applications, we exhibit a real sextic with a total of $318$ real bitangents and 6 complex ones, and perform asymptotical constructions that give the best, to our knowledge, number of real bitangents of real algebraic plane curves of a given degree.

Figures (19)

• Figure 1: Number of real bitangents to real quartic curves in terms of their real scheme.
• Figure 2: Is the number $t_s$ sharp for this isotopy type?
• Figure 3: Real schemes realized by real plane sextics.
• Figure 4: Real scheme of Harnack curves of degree $2k$ in $\mathbb{R}P^2$.
• Figure 5: Stratification of $\mathbb{R} V$.

Theorems & Definitions (54)

• Theorem 1.2
• Corollary 1.3
• Example 1.4: The case $d=4$
• Example 1.5: The case $d=6$
• Proposition 1.6
• Remark 1.7
• Theorem
• Theorem 1.8
• Remark 1.9
• Remark 1.10