Bertini Involution Demo

The Bertini involution sends the red point to the blue point.

The red curve above is the fixed point locus. When P is on this curve, Q = P. It is degree 9 and has a triple point at the eight base points not including (0,0,1). It has genus 4 = 0.5(9-1)(9-2) - 8(0.5)(3)(3-1)

The blue curve above is the image of a line under the bertini involution. It is a degree 17 curve with a singularity of order 6 at each assigned base point (not including B).

The black curve above is the curve of R throughout the pencil. Notice that B is always on the black curve, meaning that B is a flex point on some fiber of the curve. It is degree 4 and has a triple point at (0,0,1) and goes through every other base point once. It has self-intersection -1 and it is rational because it is parametrized by the line that parametrizes the whole pencil.

The orange x above is the image of P (and Q) under the double plane model of the map. The functions d^2, dw, w^2, and phi induce a map to the cone x_0x_2-x_1^2 in P^3. We project from the point (1,1,1,1) on this cone onto the plane x_1=0 to get the double plane model.