algebraic geometry

# Mori’s Bend and Break

I noticed that recently people were clicking a lot of the links I had on my blogroll. Since many of the blogs were defunct, or I didn’t read them anymore I chopped a lot off. I also added a few that I found myself returning to frequently (including no longer active ones). So that has been updated for the first time in years.

The last little bit we’ll do that is related to moduli spaces and deformation theory is something called Mori’s bend and break argument. It says that if ${X}$ is a nonsingular projective variety of dimension ${n}$ over an algebraically closed field, ${k}$, of positive characteristic ${p}$ and if there is an irreducible curve ${C\subset X}$ with ${C. K_X<0}$, then ${X}$ contains a rational curve. In this context a rational curve is an integral curve whose normalization is ${\mathbb{P}^1}$. The condition on ${K_X}$ is sometimes stated as being not numerically effective (not nef).

Suppose ${C_0}$ is an integral curve such that ${C_0. K_X<0}$. If the normalization ${C_1\rightarrow C_0}$ has genus ${0}$, then we are done. Let ${g:=g(C_1)}$. Choose ${r}$ large enough so that ${-p^r(C_0. K_X)\geq ng+1}$. Define ${q=p^r}$. Let ${F:C\rightarrow C_1}$ be the ${q}$-th power, ${k}$-linear Frobenius map and denote ${f:C\rightarrow C_0}$ the composition. We only changed the structure sheaf and not the topological space, so ${C}$ still has genus ${g}$.

Fix some point ${P\in C}$. Let ${Hom_P(C,X)}$ be the quasi-projective scheme that represents the functor of families of maps from ${C}$ to ${X}$ that keep the image of ${P}$ fixed (it's a subscheme of the usual Hom scheme). Standard deformation theory tells us that the tangent space is ${H^0(C, f^*T_X(-P))}$ and the obstruction space is ${H^1(C, f^*T_X(-P))}$.

We didn't do this, but it is all part of the package that we've talked about. When the functor is representable we get the natural estimate that ${\dim Hom_P(C, X)\geq h^0(f^*T_X(-P))-h^1(f^*T_X(-P))}$. This just comes from the fact that if every possible obstruction is realized, then each one will cut the dimension down, but often despite an obstruction space being non-zero the obstruction itself might vanish. This will only make the dimension bigger.

Now Riemann-Roch gives us that ${\dim Hom_P(C, X)\geq -q(C_0. K_X)-n+n(1-g)\geq 1}$ by our choice of ${q}$. In particular, we can find a nonsingular curve ${D}$ and a morphism ${g:C\times D\rightarrow X}$, thought of as a nonconstant family of maps all sending ${P}$ to the same point ${P_0}$. You can argue here that ${D}$ cannot be complete, otherwise the family would have to be constant.

So let ${D\subset \overline{D}}$ be a completion where ${\overline{D}}$ is a nonsingular projective curve. Let ${G:C\times \overline{D} \dashrightarrow X}$ be the rational map. Blow up a finite number of points to resolve the undefined points to get ${Y\rightarrow C\times \overline{D}}$ whose composition given by ${\pi: Y\rightarrow X}$ is an honest morphism. Let ${E\subset Y}$ be the exceptional curve of the last blow up needed. Since it was actually needed, it can't be collapsed to a point, and hence ${\pi(E)}$ is our desired curve.

This is one of those interesting things where it is easier in positive characteristic than characteristic ${0}$ because you have the Frobenius at your disposal. It allowed us to jack up the tangent space without affecting the obstruction space to produce our curve. Mori actually does relate this back to varieties in characteristic ${0}$ to prove Hartshorne's conjecture which says that a nonsingular, projective variety with ample tange bundle is isomorphic to ${\mathbb{P}^n}$.