Lefschetz bifibrations

Enter an equation for an affine surface and a linear fibration π; then a linear fibration ρ on its fibres.

Surface and π

Write powers as **, the imaginary unit as I. Decimals are read exactly: 0.3 means 3/10.

ρcompute π first

Matching pathscompute ρ first

Tracing ρ's critical values over the fibres of π is much the most expensive step, so it is asked for separately — settle on a ρ first.

Follows the solutions along the path instead of re-solving at every step. Much quicker at high degree, where the exact route is not practical; leave it off for small surfaces, where the exact route is instant and exact. Only means anything when the paths are tracked: it is the tracking it checks. The strongest check there is, but that one solve can cost many times the tracking -- about twenty times, on a degree-14 surface -- so it is off unless you ask for it.

Additional computationscompute ρ first

Paths that are not vanishing paths: any route through either base, with both ends free to move. Draw one, press Compute, and the result appears beside it — read the same way as above, so an open path reports the two branches that meet at its end and a closed loop the permutation it induces.