Combinatorial conditions for linear systems of projective hypersurfaces

Miguel Marco

Santiago de Compostela, October 8th 2016

Before starting

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Survey of past and current work

Partly joint with J.I. Cogolludo

Two points determine a line.

\(a_{0,0}z+a_{1,0} x + a_{0,1}y\)

One point determines a pencil of lines

\(a_{0,0}z+a_{1,0} x + a_{0,1}y\)

Five (generic) points determine a conic.

\(a_{0,0}z^2+a_{1,0} xz + a_{0,1} yz +a_{2,0} x^ 2+a_{1,1}x y +a_{0,2}y^2\)

Four points determine a pencil of conics

\(a_{0,0}z^2+a_{1,0} xz + a_{0,1} yz +a_{2,0} x^ 2+a_{1,1}x y +a_{0,2}y^2\)

Four points determine a pencil of conics

\(a_{0,0}z^2+a_{1,0} xz + a_{0,1} yz +a_{2,0} x^ 2+a_{1,1}x y +a_{0,2}y^2\)

Nine points determine a cubic

\(a_{0,0}z^3+a_{1,0} x z^2+ a_{0,1} yz^2 +a_{2,0} x^ 2z+a_{1,1}x yz +a_{0,2}y^2z+a_{3,0}x^3+a_{2,1}x^2y+a_{1,2}xy^2+a_{0,3}y^3\)

Eight points determine a pencil of cubics

\(a_{0,0}z^3+a_{1,0} x z^2+ a_{0,1} yz^2 +a_{2,0} x^ 2z+a_{1,1}x yz +a_{0,2}y^2z+a_{3,0}x^3+a_{2,1}x^2y+a_{1,2}xy^2+a_{0,3}y^3\)

Eight points determine a pencil of cubics?

\(a_{0,0}z^3+a_{1,0} x z^2+ a_{0,1} yz^2 +a_{2,0} x^ 2z+a_{1,1}x yz +a_{0,2}y^2z+a_{3,0}x^3+a_{2,1}x^2y+a_{1,2}xy^2+a_{0,3}y^3\)

Theorem (Cayley-Bacharach): - If a cubic passes through eight intersections points of two other cubics, it also passes through the ninth.

Question:

When is there a pencil of curves?:

Two curves always generate a pencil

More than three are in a pencil iff every three of them are.

So the question really is:

What conditions should three curves satisfy to be in a pencil?

Not as trivial as it might seem:

First result

Noether fundamental theorem (\(af+bg\)):

Let \(f,g,h\) be three homogenous polynomials in \(\mathbb{C}[x,y,z]\). Being \(f\) and \(g\) coprime. Then \(h\in (f,g)\) if and only if \(h\in (f,g)_p \forall p\in \mathbb{P}^2\). Where \((f,g)_p\) is the ideal generated by \(f\) and \(g\) in the localization of \(\mathbb{C}[x,y,z]\) in the maximal ideal corresponding to \(p\).

Remarks about the Fundamentalsatz

The condition is trivially satisfied for the non-intersection points.

It reduces the problem to studying the problem locally at the base points.

Definition: Given a line arrangement \(\mathcal{L}=\{l_0,\ldots, l_n\}\), a partition \(\mathcal{L} = \mathcal{L}_0 \coprod \mathcal{L}_1 \coprod \cdots \coprod\mathcal{L}_m\) and an exponent function \(d:\mathcal{L}\mapsto \mathbb{Z}^+\) is said to form a combinatorial pencil if at each intersection point, one of these conditions hold:

The point is "monochromatic" (all the lines lie in the same \(\mathcal{L}_i\))

All the components have the same multiplicity in the point (\(\sum_{l\in\mathcal{L}_i}d(l) = \sum_{l\in\mathcal{L}_j}d(l)\))

Theorem (Falk-Yuzvinsky, M.): The previous method gives all (primitive) combinatorial pencils.

Theorem (Falk-Yuzvinsky): A line arrangement is a union of three or more curves in a pencil if and only if it admits a combinatorial pencil.

Example:

Example:

Generalization: arbitrary curves

Different types of singularities:

Algebraic invariants of a singularity

Definition: Given a local branch \(f=0\) at the origin, its multiplicity is defined as the maximum \(p\) for which \(f\in \mathfrak{m}^p\)

Definition: Given two local branches \(f=0,g=0\) at the origin, its intersection multiplicty is defined as \[dim\frac{\mathcal{O}_p}{(f,g)}\]

Tool for understanding singularities: blowup

Locally:

Consider a point \(p\).

The set of lines through \(p\) form a \(\mathbb{P}^1\).

We have a map \(\pi:\mathbb{L}\times\mathbb{P}^1\mapsto \mathbb{A}^2\)

\(\pi\) is bijective outside of \(p\)

\(\pi^{-1}(p)=\mathbb{P}^1\)

Properties of the blowup

The blowup "smoothens" the singularity.

Theorem: Every plane curve can be resolved to a normal crossing divisor by a finite number of blowup at points.

Lemma: Let \(f,g\) be local branches in a point \(p\). Let \(\bar{f},\bar{g}\) be their strict transforms. Then \[ (\bar{f},\bar{g}) = (f,g) - m_p(f)\cdot m_p(g)\]

Generalization of the incidence lattice

After resolving to normal crossings, we have:

The strict transforms of the original components.

The exceptional divisors that appeared during the blowup process

An incidence between them given by the multiplicity

Generalization of the process

Choose base exceptional divisors

Construct the incidence matrix

Proceed as before

Example

\[
(- x^{3} + y^{3} - y^{2}) \cdot x \cdot (y - 1) \cdot y
\]

Definition: Given a plane curve, with irreducible components \(\mathcal{C}=\{C_1,\ldots C_m\}\) a combinatorial pencil is a partition \(\mathcal{C}=\mathcal{C}_1\coprod\mathcal{C}_2\coprod \cdots \coprod \mathcal{C}_n\) and an exponent function \(d:\mathcal{C}\mapsto \mathbb{Z}^+\) such that at any singular point \(p\) of the curve, one of the following two options hold:

All components that go through \(p\) live in the same \(\mathcal{C_i}\)

For each local branch \(b\) at \(p\), \(b\in \mathcal{C}_i\), and every \(j,k\neq i\), the following equality holds:

Theorem (Cogolludo, M.): A Curve allows a combinatorial pencil if and only if it is the union of three or more fibers of a pencil of curves. Moreover the fibers are given by the partition and exponents of the combinatorial pencil.

Higher dimensions

Question: Given \(n+1\) hypersurfaces in \(\mathbb{C}\mathbb{P}^n\), when do they belong to a linear system of dimension \(n-1\)?

Still open

Partial answer

Definition (Libgober): A hyperplane arrangement is said to be an isolated non normal crossing (INNC) if the intersection of \(i\) hyperplanes has codimension \(i\) except maybe in isolated points.

Theorem (M.): An INNC arrangement in \(\mathbb{C}\mathbb{P}^n\) where at most \(n+1\) hyperplanes meet at a point is a union of \(n+1\) fibers of a linear system of dimension \(n\) if and only if it admits a partition such that every \(INNC\) point is the intersection of one hyperplane in each element of the partition.