Maximally Inflected Curves: 3

Visualizing Linear Series

Visualize a linear series via the resulting map ${\mathbb{P}}^1 \to {\mathbb{P}}^k$

$\to$ To see it, both the series and its ramification points must be real.

Definition. A ${\mathbb{R}}$ eal linear series of dimension (map $\varphi\colon{\mathbb{P}}^1_{\mathbb{R}} \to {\mathbb{P}}^k_{\mathbb{R}}$ )
of degree is maximally inflected if all of its ramification is real.

Example ( and ):

Ramification $\alpha$

flex $\iota$

cusp $\kappa$

planar

	Definition. A ${\mathbb{R}}$ eal linear series of dimension (map $\varphi\colon{\mathbb{P}}^1_{\mathbb{R}} \to {\mathbb{P}}^k_{\mathbb{R}}$ )
	of degree is maximally inflected if all of its ramification is real.