Genus, differential forms, and algebraic geometry

[mathwonk]

i made a mistake about the use of the word genus for higher dimensional varieties - usually the words geometric genus are used not for the number of 1 forms, but for the number of n forms, where n = dim of variety.

 

[mathwonk]

another rmark to hurkyl on genera of plane quartics. a plane quartic like y^2 = quartic in x, has arithmetic genus 3 but geometric genus 1.

i.e. this is a singular curve and there are two ways to associate a non singualr curve to it: one is to vary the equatioin a little, i.e. the coefficients, until the curve becomes non singular. the genus of that non singular curve is the arithmetic genus of the original curve.

the other way is to "desingularize" the curve, by removing all singular points, creating 2 punctures, and then putting in a disc centered at each puncture.

i.e. topologically i believe this curve looks like a modified curve of genus one, i.e. a torus but with two distinct points identified. Then one can do surgery in several ways on this curve.

start by removing a small nbhd of the singular point, i.e. of the point resulting from gluing two points of the torus together. That leasves two disc shaped holes.

then the simplest thing to do is glue in two discs, one in each hole, giving sa torus, of genus one.

but one can also glue in any other manifold with boundary whose boundary consists of exactly two discs, such as a cylinder giuving a curve of genus 2, or a curve of genus one with two discs removed, giving a curve of genus three.

this last is what happens when we simply vary the coefficients until the curve becomes a smooth quartic.

one can see this dynamically, by varying the equation of a smooth quartic as follows:

consider y^2 - ey^3 = x(x^3-1), and let e go to zero. as e goes to zero, the torus with two discs removed collapses into the singualr point. the lost homology cycles are called "vanishing cycles" and lefschetz studied them deeply in pencils of varying surfaces..

deligne used lefschetz pencils to prove the weil conjectures 30 years ago.

 

[mathwonk]

here is my favorite way to calculate the genus of a plane cubic: note that a cubic degenerates to a triangle, anfd a triangle ahs one hole, so the genus is the number of holes namel;y one. for, a quartic note there are three holes in a triangle (with infinijtely long sides) plus one line.

what is the genus of a quntic?