Trazabilidad de las muestras
MARCO REFERENCIAL
Definition 3.1.1. Given an arrangementAwith defining equationQ=f1· · ·fn and
a choice of weight vector λ = (λ1, . . . , λn)∈ Cn, the corresponding master function is defined by
Φλ=f1λ1· · ·fλn n .
A point x ∈ C` is called a critical point with respect to weight λ if it is a critical point of the function Φλ, i.e.
∂Φλ
∂xi |x= 0, for i= 1, . . . , `.
Note that regardless of the terminology, a master function is not actually a func- tion, as it is multi-valued. However, by the following lemma, if we restrict to points where the C` factor is in the complement of the hyperplanes, we get a description as
the vanishing of the differential of a logarithmic function, which is a (well-defined) meromorphic 1-form.
Lemma 3.1.2. A point x∈M(A) is critical with respect to a weightλ if and only if it is a root of the differential 1-from
ωλ =
n X i=1
λidfi/fi = d log Φλ. (3.2)
Proof. A point x∈M(A) is critical if it is a root of
∂Φ/∂xj = Φ
n X i=1
λi(∂fi/∂xj)fi−1
for allj = 1, . . . , `. Since the value of Φ is nonzero, these equations can be compactly written as the vanishing of the single formula
` X j=1 n X i=1 λi(∂fi/∂xj)fi−1dxj = 0
which recoversωλby just flipping the summations. See [23] for a slightly more general treatment.
Definition 3.1.3. Let C denote the coordinate ring of the space of weights Cn, namely C[a1, . . . , an]. Define the logarithmic 1-form by
ωa = n X i=1 ai dfi fi .
It is immediate to see from the definitions that ωa ∈ Ω1(A)⊗CC =: Ω1C(A). The
vanishing of this one form is defined by an ideal, called the meromorphic ideal Imer,
which is generated by ` rational polynomials, namely h∂xj, ωai, forj = 1, . . . , `.
Let us save the letter S for the ring R⊗CC =C[x1, . . . , x`;a1, . . . , an]. Note
that Ω1C(A) is an S-module. Note that the meromorphic idealImer lives in the local
ring SQ.
Definition 3.1.4. Given an arrangementA={H1, . . . , Hn}, together with a surjec-
tive map µ: {a1, . . . , am} → A for some m ≥ n, one naturally gets a multiarrange-
3.1. CRITICAL POINTS above, let ωµ = n X i=1 X j∈µ−1(Hi) aj dfi fi . In simple case, n=m and µ:ai 7→Hi, for i= 1, . . . , n.
Multirestriction in the sense of Ziegler provides a natural example of the above construction as follows. LetA be a simple arrangement of sizenwith a distinguished hyperplane H0. Then define a map
z :{a1, . . . , an−1} → A00
by assigning ai 7→ K ∈ A00, if Hi∩H0 = K. Note that |(A00, z)| = n−1, where
z :A00 →Nas a multiplicity is defined in Formula1.13. Under the notation just intro- duced, we have the following Proposition, which is a modification of [23, Proposition 4.1].
Proposition 3.1.5. Let (A,m) be a multiarrangement, then the zero locus of ωµ is a nonsingular quasi-affine variety, denoted Σµ(A), whose codimension in C`×Cm is rank (A).
Proof. Without loss of generality, we may assume that A is full rank. Let fi(x) = P`
j=1cjixj. The solution space of ωµ = 0 is the common zero set of
dµ,i(x) = n X i=1 ( X j∈µ−1(Hi) aj) cji fi(x) ,
for j = 1, . . . , `, with x ∈ C` and a ∈ Cm, where m =|m| and n = |A|. In matrix notation, this is the same as the solution space of
h cji/fi(x) i P j∈µ−1(H1)aj .. . P j∈µ−1(Hn)aj = 0,
where we think of x as a parameter that runs over all points of the complement, a space of dimension`. Having picked an x∈M(A), the linear system above admits a solution space of dimensionn−`. Again after fixing every solution, we get a space of dimensionm(Hi)−1, for theithcomponent. Adding up the dimensions returnsm−`. Therefore the solution space in C`×Cn is of dimension m after the contribution of
x∈M(A) is added, hence of codimension `. Smoothness is implicitly verified in the above argument, as the dimension of the tangent space is the same at all points. The above argument also verifies the fact the natural projection Σµ → M(A) is a trivial
vector bundle.
Definition 3.1.6. Let A be a simple arrangement. The logarithmic ideal I(A) (as introduced in [9]) is the image of the module of C-linear derivations DC(A) := D(A)⊗CC (asS-module) and ωa under the pairing of Proposition 1.4.22 for p= 1.
I(A) :=hDC(A), ωai (3.3)
Proposition 3.1.7. Let (A,m) be a multiarrangement. The closure of Σµ is defined by the idealIµ(A) = hDC(A), ωµi, which is bi-homogeneous with respect to the grading of R and C.
Proof. Similar to [9, Theorem 2.9].
Proposition 3.1.8. Under the above notations, the intersectionV(Iµ(A))∩(M(A)×
Cm) equals Σµ.
Proof. This follows from Lemma 1.4.27. See [23] for a proof in the simple case.
3.1.1
A Concrete Example
One of the key results of this chapter is Theorem 3.2.2 which relates the logarithmic ideal of an arrangement to that of its deletion and restriction. Here is an example to illustrate the answer.
Consider the rank 2 braid arrangementA, defined by
Q=xy(x−y)
The logarithmic ideal I(A) lives in the polynomial ring R⊗CC, where R =C[x, y] and C = [a1, a2, a3]. In an algebraic language, the main result of this chapter is
centered around understanding the idealI(A) + (ai), for any variableai. It turns out that the combinatorics of A emerges in the associated primes of this ideal.
We know that A is free with a basis consisting of the Euler derivation θE = x∂x+y∂y and θ =xy(∂x+∂y) [use Saito’s criterion, Theorem 1.4.14]. Pairing these
derivations with the meromorphic one form
ωa =a1 dx x +a2 dy y +a3 d(x−y) x−y