# Polishchuk Chap. 17 constructions and questions

Polischuk claims if is smooth and projective then any object in (bounded derived category of coherent sheaves) is isomorphic (in the derived sense) to a finite complex of locally free sheaves. This is roughly the content of III.6.9 in Hartshorne, this problem requires to be Noetherian, integral, separated and regular.

Being in this situation means you can take determinants. For a locally free sheaf its the top exterior power. For a finite complex of locally free sheaves this is the alternating tensor power of the determinants of each of the pieces.

see this related post regarding the red parts.

### Determinant, section construction

is a relatively smooth projective curves. I guess this means is a curve, and is smooth (of relative dimension zero?; see vakil…), and assume is flat over S and (where does this come into play (think it might show rk )?

Do: such that are flat over , (what’s the reason flatness keeps coming into the picture). Assuming (apparently can take for a relatively effective divisor) should imply (how? ; long exact sequence does it ; dim = 1) .

Also that should say are vectors bundles, (cohomology with base change, would want to be constant, comes from flatness?, or how does this work?)

One important thing I didn’t immediately realize is that is a curve! So the only potential higher cohomology is .

In any case, these claims are meant to justify that in the bounded derived category. Does this use some spectral sequence magic, or something like E_i are pi_* acyclic?

In any case, is well defined. Somehow , but , so the pushforwards of have the same rank. So the map is represented by a sheaf version of a square matrix, hence has a sheaf version of a determinant. So Polishchuk says should be a section of , but it seems is just a section of . A section of should just be an alternating map , for example taking n vectors in and taking the determinant of the matrix they form. So can describe sections of each peice individually, but what is section of the product?

Then there is a nontrivial argument that shows this construction doesn’t so much depend on the resolution . But I didn’t really work through this.

### Curve Mapping to an Abelian Variety construction

The setup

- map from curve to abel. var.
- cohrent, .
- lands in the bounded derived category so is defined.

Certainly for a a line bundle on there is s.e.s (this is by construction exact when localized at every point)

Some general derived category nonsense says the natural functor from is exact meaning s.e.s get sent to distinguished triangles so there is something like an exact triangle

If you just write everything out (see prop. 17.1 of Polishchuk) then what you want happens:

The magical thing is that this means doesn’t depend on the rational equvivalence class of so there is a well defined map

Facts

- is the composition . (note the latter is the map .
- is a homo of var then .

### Principal Polarization of Jacobian

Given a curve with a point then it has a Jacobian and a line bundle (normalized at p) on . By the universal property of the Poincare bundle on , there is such that

maps:

- is the identity (a tautology).
- This implies
- sending satisfies (see below)

By universal property of there morphism is equivalent to a line bundle on . In fact . Tautologically the map is the identity. Putting universal properties together, is actually the pullback of along

So this gives statement 3.

The magical thing that happens is is equal to . But . So we get an inverse in one direction, but what about the other direction?

## 1 Comment

Jump to comment form | comment rss [?] | trackback uri [?]