*Bigness*

Throughout this talk, is a field of characteristic zero, algebraically closed unless otherwise specified.

A **variety** is an integral scheme, separated and of finite type over
a field.

Throughout this talk, is a complete variety over .

**Definition: ** Let
be a line sheaf on . We say that
is
**big** if there is a constant such that
for all sufficiently large
and divisible
.

**Lemma. ** (Kodaira) Let
be a line sheaf and
an ample line
sheaf on . Then
is big iff
has a (nonzero) global section
for some .

*Proof: * ``
'' is obvious.

``'': Write with a reduced effective very ample divisor. It will suffice to show that has a global section for some . Consider the exact sequence

**Definition: ** A vector sheaf
of rank on is **big** if there is
a such that

Equivalently, is big iff on is big.

*Essential base locus*

**Definition: ** Assume that is projective, and let
be a (big)
line sheaf on . The **essential base locus** of
is the subset

**Question: ** If
is a big vector sheaf, is its essential base locus
properly contained in ?

**Answer: ** No. Example: Unstable
over curves.

**Question: ** What if
is big and semistable?

*Curves*

Throughout this section, is a (projective) curve.

**Definition: ** (Mumford) A vector sheaf
on is **semistable** if,
for all short exact sequences

**Theorem. ** Let
be a big semistable vector sheaf on . Then
is ample (i.e.,
is ample on
). In particular,
the essential base locus of
is empty.

*Proof: * By Kleiman's criterion for ampleness, the sum of an ample and
a nef divisor is again ample, so by Kodaira's lemma it suffices to show
that if
is a semistable vector sheaf on , then all effective
divisors on
are nef.

So, let be an effective divisor and a curve on . We want to show:

Since is semistable, so is (proof later).

Therefore we may assume that is a section of , and that is a prime divisor.

Since is a section, it corresponds to a surjection . Moreover, . By semistability, therefore,

Now consider . Let be the degree of on fibers of ; . Then for some . Thus corresponds to a section of , hence we have an injection

Since is semistable, so is (proof later); hence

Let ; then has rank . The diagram

and therefore by (**),

*Higher Dimensional Varieties*

Let again be a complete variety of arbitrary dimension.

**Construction: ** Given a vector sheaf
on of rank and a representation

Examples of this include , , and .

**Definition: ** (Bogomolov) A vector sheaf
of rank on is
**unstable** if there exists a representation
of determinant 1 (i.e., factoring through ) such that
has a nonzero section that vanishes at at least one point.
It is **semistable** if it is not unstable.

**Theorem. ** (Bogomolov) If is a curve, then Bogomolov's definition of
semistability agrees with Mumford's.

**Remark: ** If has determinant then
,
but not conversely.

Indeed, the representation , , has image contained in but its does not factor through .

To see that the (true) converse holds, first show that the vanishing of the determinant defines an irreducible subset of ; this is left as an exercise for the reader. Now suppose that is a representation that factors through , and suppose also that its image is not contained in . Then is a nonconstant regular function , hence it determines a nonconstant rational function on with zeros and poles contained in . But the latter is irreducible, so it can't have both zeroes and poles there, contradiction.

So now we can pose:

**Question: ** If is a projective variety and
is a big, semistable
vector sheaf on , then is the essential base locus of
a proper
subset of ?

**Remark: ** We can't conclude that
is ample in the above, as the following
example illustrates. Let be a projective variety of dimension ,
let
be a big semistable vector sheaf on of rank ,
let
be the blowing-up of at a closed point, and let
be the exceptional divisor. Then the essential base locus of
must contain .

*My Mitteljahrentraum
*

The question of an essential base locus being a proper subset comes up in Nevanlinna theory, and I hope to be able to use it in number theory, as well. Here's how.

Bogomolov has shown that is semistable for a smooth surface . One would hope to generalize this, to for a normal crossings divisor on , and also to higher dimensions. Then it would suffice to prove that one of these bundles is big to get arithmetical consequences.

Moreover, Bogomolov's definition of semistability can be generalized to defining semistability of higher jet bundles. These are not vector bundles, because they correspond to elements of for a group other than . But, one can make the same definition, using those representations of having the appropriate kernel: again (Green-Griffiths), or a certain bigger group (Semple-Demailly). Probably the latter.

Bigness is easy to define in this context, and then one hopefully can use the two properties to talk about the exceptional base locus. Already the proof of Bloch's theorem in Nevanlinna theory can probably be recast in this mold.

*Is Semistability Really Necessary?*

The proof of the main theorem of this talk didn't really need the full definition of semistability; it only used the condition on the degrees of subbundles for subbundles of rank 1 and corank 1. Would the following definition make sense, and would it be preserved under pull-back and symmetric power?

**Definition: ** Let be a projective curve and let
be a vector sheaf
of rank on . Then
is **-semistable** if
the condition on degrees and ranks of subbundles holds for all full subbundles
of rank and corank .

Again, what would be a reasonable representation-theoretic formulation of this definition?

*Loose Ends*

In the proof of the main theorem it remains to show that semistability is preserved under pull-back and under taking .

To show the first assertion, let be generically finite, and let be a semistable vector sheaf on . Suppose that is unstable. Let be a representation such that has a nonzero global section that vanishes somewhere. Let . Then taking norms gives a global section of

The second assertion is proved similarly: suppose there is a representation

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for Rational and integral points on higher dimensional varieties.

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