Logarithmic Discriminants of Hyperplane Arrangements
Overview
Abstract. A recurring task in particle physics and statistics is to compute the complex critical points of a product of powers of affine-linear functions. The logarithmic discriminant characterizes exponents for which such a function has a degenerate critical point in the corresponding hyperplane arrangement complement. We study properties of this discriminant, exploiting its connection with the Hurwitz form of a reciprocal linear space.
On this page you can find the Macaulay2 and Julia code to compute the examples in the paper, as well as the computer proofs of Lemma 5.3 and Corollary 5.5.
Computing logarithmic and Hurwitz discriminants
The logarithmic discriminant and the Hurwitz discriminant can be computed using variable elimination, at least in small cases.
We implemented this in Macaulay2, the code can be downloaded here. LogDisc.m2
To verify Example 1.1 (here in the notation of Example 8.1) and Example 3.3 you can run the following:
load "logdisc.m2"
(M,S) = M0nArrangement 5
elapsedTime( nabla = logDisc(M, PolyRing=>S); )
factor (nabla_0 + 4*s_(1,3)*s_(1,4)*s_(2,3)*s_(2,4))
bA = matrix {{1,1,1,0},{2,1,3/2,0},{1,2,3/2,0},{0,1,0,1},{0,0,1,1},{0,1,1,2}}
elapsedTime( nabla = logDisc bA; )
decompose nabla
The first computation terminates within a second, while the second takes about 25 seconds on our system. We warn that this code will likely not terminate with larger numbers of hyperplanes. Already the logarithmic discriminant of the \(\mathcal{M}_{0,6}\)-arrangement seems to be far out of reach (it is of degree \(30\) in \(9\) variables, see below).
Birationality of the projection for d=1
In order to prove the claim in Lemma 5.3, we need to show that for \(n+1 \geq 3\) distinct points the linear system
has full rank. In the case of four forms \(\{x,x+1,x+b_2,x+b_3\}\) the coefficient matrix is the following:
We can compute and factorize its determinant symbolically in Macaulay2:
R = QQ[x,y,b_2..b_3]
bb = {0,1,b_2,b_3}
C4 = transpose matrix for i from 0 to 3 list {1/(x+bb_i), 1/(x+bb_i)^2, 1/(y+bb_i), 1/(y+bb_i)^2}
factor det C4
If the numbers \(\mathcal{A} = \{0, -1, -b_2, -b_3\}\) are distinct and \(x,y \in \mathbb{C}^1 \setminus \mathcal{A}\) are also distinct, then \(C_4\) has full rank as desired. For any number of \(n+1\) points the corresponding system contains \(C_4\) (or is contained, in the case \(n+1=3\)), hence has full rank.
This shows that \(\mathcal{B}\) is indeed a trivial projective bundle over \(\{\,(x,y) \in X\times X \mid x\neq y\,\}\) if \(n+1\geq 5\) and empty if \(n+1\leq 4\).
The discriminant of four points is always singular
To verify the claim in Corollary 5.5, we can compute the discriminant (in the sense of multivariate polynomials) of the logarithmic discriminant for the generic arrangement over \(\mathbb{Q}(b_2,b_3)\). We use the formula from Corollary 5.4 to compute \(\Delta_{\log}\) for generic points \(\mathcal{A} = \{0, -1, -b_2, -b_3\}\). The discriminant of \(\Delta_{\log} \in \mathbb{Q}[b_2,b_3][u_0,\dots,u_3]\) is a polynomial \(\operatorname{Disc}_u \Delta_{\log} \in \mathbb{Q}[b_2,b_3]\) which vanishes for \((b_2,b_3) \in \mathbb{C}^2\) if any only if \(\nabla_{\log}\) is singular.
needsPackage "Resultants"
R = QQ[b_2..b_3][u_0..u_3][x_0..x_1]
bb = {0,1,b_2,b_3}
g1 = sum(4, i -> u_i * product( toList(set(0..3)-set({i})), k -> x_1 + bb_k*x_0 ))
logDisc = discriminant g1
discriminant logDisc
The result is identically zero.
Computing the logarithmic discriminant numerically
Conjecture 1 states that the degrees of the logarithmic discriminants of the moduli space \(\mathcal{M}_{0,m}\)
for \(m = 5, 6, 7, 8\) are given by \(m = 4, 30, 208, 1540\) respectively. This can be verified numerically using
this Julia script. NumericLocDisc.zip
The surface \(\mathcal{M}_{0,5}\) can be modeled as the complement of five lines in \(\mathbb{C}^2\).
The Hurwitz form of the associated reciprocal linear space has degree eight. We computed this using finite
field methods and numerical interpolation. The resulting polynomial can be downloaded here: HurwitzForm_M05.txt