# Incidence structures on algebraic curves

Posted on June 7, 2016 by Dima

tags: incidence structures, zariski geometries

## Abstract projective geometry

Let $$k$$ be a field. Consider the set of points $$P$$ and lines $$L$$ on the projective $$n$$-space $${\operatorname{\mathbb{P}}}^n(k)$$ over $$k$$. One finds that the following properties hold:

• there is exactly one line incident to any two distinct points
• there is exactly one point that is incident to two distinct lines
• every line contains at least three distinct points

A tuple $$(P,L,I)$$ where $$P$$ is the set of points, $$L$$ is the set of lines and $$I \subset P \times L$$ is the incidence relation, is an abstract projective geometry if the properties above hold. These properties in particular hold if $$P$$, $$L$$ are sets of points and lines on a projective plane over a (skew) field $$k$$. There are examples of abstract projective geometries that do not arise this way. The reason is that all geometries arising from a (skew) field additionally satisfy Desargues axiom:

Given two triangles $$ABC$$ and $$A'B'C'$$ as above, the lines $$AB$$ and $$A'B'$$, $$BC$$ and $$B'C'$$, and $$AC$$ and $$A'C'$$ intersect in three points that lie on the same line.

There are two affine statements that follow from this axioms: called small and big Desargues axioms.

small Desargues: given $$A$$ and $$A'$$, $$B$$ and $$B'$$, $$C$$ and $$C'$$ lying on three parallel lines, $$BC$$ and $$B'C'$$ are parallel.

big Desargues: given that the lines $$AB$$ and $$A'B'$$, and $$AC$$ and $$A'C'$$ are pairwise parallel, the lines $$BC$$ and $$B'C'$$ are parallel.

If $$k$$ is commutative, then a Pappus axiom is also satisfied:

# ddc lemma

Posted on May 4, 2016 by Dima

tags: kahler manifolds, ddc

In this note I will prove the easy local case of $$dd^c$$ lemma.

Let $$(M,I)$$ be a complex manifold. Extend the action of the complex structure on exterior powers of the complexified tangent bundle forms by $\mathbf{I}: \bigwedge{}^* M \to \bigwedge{}^* M, \mathbf{I} := \sum i^{p-q} \cdot \Pi^{p,q}$ where $$\Pi^{p,q}: \bigwedge^{p+q} \to \bigwedge^{p,q}$$ is the natural projection. Define a dwisted differential $$d^c = I \circ d \circ I^{-1}$$. On the cotangent bundle define it to be $$\mathbf{I}(\alpha)(v_1, \ldots, v_n) := \alpha(\mathbf{I}v_1, \ldots, \mathbf{I}v_n)$$.

Proposition. $$\partial = \dfrac{d + i \cdot d^c}{2}, \bar \partial = \dfrac{d - i \cdot d^c}{2}$$, where $$\partial$$ and $$\bar\partial$$ are projections of $$d$$ on $$(\cdot +1, \cdot)$$ and $$(\cdot, \cdot+1)$$ components.

Proof. Indeed $\partial + \bar\partial + i \mathbf{I}^{-1} (\partial + \bar\partial) \mathbf{I} = \partial + \bar\partial + i (i^{q-p-1} \partial + i^{q-p+1}\bar\partial)i^{q-p}=\\ = \partial + \bar\partial + \partial + i^2\bar\partial=2\partial$ Similarly for the second equality. $$\square$$

In particular, $$\partial\bar\partial = -\dfrac{i}{2}dd^c$$ on a complex manifold.

Lemma (Poincaré lemma). If $$\alpha$$ is a closed form on a polydisc then it is exact.

Lemma (Poincaré-Dolbeault-Grothendieck lemma). If $$\alpha$$ is a $$\partial$$-closed, not holomorphic (i.e. $$\alpha \notin A^{n,0} M$$), form on a polydisc then it is $$\bar\partial$$-exact.

(this lemma means in particular that Dolbault resoltions of sheaves of holomorphic forms are acyclic)

Note that since $$\bar(\partial\alpha) = \bar\partial \bar\alpha$$, the PDG lemma also holds for $$\partial$$.

Lemma (local $$dd^c$$ lemma). Let $$\eta$$ be a $$(1,1)$$-form on a polydisc. Assume either

• $$\eta$$ is exact, or
• $$\eta$$ is $$\partial$$-exact, $$\bar\partial$$-closed.

Then there exists a function $$\chi$$ such that $$\eta = dd^c \chi$$.

Proof. First establish the equivalence of the assumptions. Recall that $$d^2=\partial^2=\bar\partial^2=0$$, where $$d=\partial+\bar\partial$$, and therefore $$\bar\partial\partial=-\partial\bar\partial$$. If $$\eta = d\alpha$$ then there exist forms $$\beta \in A^{1,0}, \gamma\in A^{0,1}$$ such that $$\eta = \partial\beta + \bar\partial\gamma$$ and $$\bar\partial\beta=\partial\gamma=0$$. Then $$\partial(\eta) = \partial^2 \beta + \partial\bar\partial\gamma=-\bar\partial\partial\gamma=0$$. Similarly for $$\bar\partial\eta$$. Apply PDG and $$\partial$$-PDG to get $$\partial$$ and $$\bar\partial$$-exactness of $$\eta$$.

By Poincaré-Dolbeault-Grothendieck lemma, there exists a form $$\alpha \in A^{1,0}$$ such tha $$\bar\partial\alpha=0$$. Deduce $$\bar\partial\partial \alpha = -\partial\bar\partial \alpha=-\partial \eta = 0$$.

Therefore $$\partial\alpha$$ is a closed form: $$(\partial + \bar\partial)\alpha = \partial^2 \alpha + \bar\partial\alpha=0$$. By Poincar'e lemma, there exists a form $$\alpha'$$ such that $$d\alpha' = \partial\alpha' + \bar\partial\alpha = \partial\alpha$$. By grading considerations, $$\bar\partial\alpha = 0$$.

Consider the form $$\beta = \alpha - \alpha' \in A^{1,0}$$. Then $$\partial\beta = \partial\alpha - \partial\alpha' = 0$$ and $$\bar\partial\beta = \bar\partial\alpha - \bar\partial\alpha' = \eta$$. By $$\partial$$-PDG, there exists a function $$\psi$$ such that $$\partial\psi = \beta$$, so $$\bar\partial\partial\psi=\eta$$. Put $$\chi=-i\psi$$. $$\square$$

Interestingly, this statement is true globally, i.e. for global $$(p,q)$$-forms on a manifold $$M$$, if the manifold is Kähler.

Lemma (global $$dd^c$$ lemma). Let $$\alpha$$ be a form on a Kähler manifold, and assume that $$\eta$$ is either $$d$$-, $$\partial$$- or $$\bar\partial$$-exact. Then $$\alpha$$ is $$dd^c$$-exact (or, which is the same, $$\partial\bar\partial$$-exact).

This result is not as trivial, and requires Hodge theory.

# The Picard scheme

Posted on April 24, 2016 by Dima

tags: moduli, picard

## Functors $$Pic$$ and $$Div$$

We will denote base change with a subscript: $$X_T = X \times T$$.

If $$X$$ is a scheme, the Picard group of $$X$$ is defined to be the group of isomorphism classes of invertible sheaves on $$X$$. The relative Picard functor of an $$S$$-scheme $$X$$ is defined as ${\operatorname{Pic}}_{X/S}(T) := {\operatorname{Pic}}(X_T) / {\operatorname{Pic}}(T)$ where the embedding $${\operatorname{Pic}}(T) \hookrightarrow {\operatorname{Pic}}(X_T)$$ is given by the pullback along the structure maps $$X_T \to T$$.

Caveat: all representability results work with a sheafification of this functor in some topology, Zariski, étale, or fppf. Unless $$X \to S$$ is proper and has a section, they need not be isomorphic.

An effective divisor is a closed subscheme such that its ideal is invertible. If $$f: X \to S$$ is a morphism of schemes then a relative effective divisor on $$X$$ is an effective divisor $$D$$ such that $$D$$ is flat over $$S$$. For a morphism $$X \to S$$ define the functor of relative divisors ${\operatorname{Div}}_{X/S}(T) := {\{\ \textrm{ relative effective divisors on } X_T/T \ \}}$

We are interested in representability of this functor, so to this end we prove a little lemma.

Lemma. Let $$X \to S$$ be a flat morphism. Let $$D$$ be a closed subscheme of $$X$$ flat over $$S$$. Then $$D$$ is relative effective divisor in a neighbourhood of $$x \in X$$ if and only if $$D_s$$ is cut out in a neighbourhood of $$x$$ in $$X_s$$, where $$s$$ is the image of $$x$$, by a single non-zero element of $${\operatorname{\mathcal{O}}}_{X_s,x}$$ (which amounts to being an effective divisor in $$X_s$$ but we won’t prove it).