  In 2013, Gau and Wu introduced a unitary invariant, denoted by $k(A)$, of an
$n\times n$ matrix $A$, which counts the maximal number of orthonormal vectors
$\textbf x_j$ such that the scalar products $\langle A\textbf x_j,\textbf
x_j\rangle$ lie on the boundary of the numerical range $W(A)$. We refer to
$k(A)$ as the Gau--Wu number of the matrix $A$. In this paper we take an
algebraic geometric approach and consider the effect of the singularities of
the base curve, whose dual is the boundary generating curve, to classify
$k(A)$. This continues the work of Wang and Wu classifying the Gau-Wu numbers
for $3\times 3$ matrices. Our focus on singularities is inspired by Chien and
Nakazato, who classified $W(A)$ for $4\times 4$ unitarily irreducible $A$ with
irreducible base curve according to singularities of that curve. When $A$ is a
unitarily irreducible $n\times n$ matrix, we give necessary conditions for
$k(A) = 2$, characterize $k(A) = n$, and apply these results to the case of
unitarily irreducible $4\times 4$ matrices. However, we show that knowledge of
the singularities is not sufficient to determine $k(A)$ by giving examples of
unitarily irreducible matrices whose base curves have the same types of
singularities but different $k(A)$. In addition, we extend Chien and Nakazato's
classification to consider unitarily irreducible $A$ with reducible base curve
and show that we can find corresponding matrices with identical base curve but
different $k(A)$. Finally, we use the recently-proved Lax Conjecture to give a
new proof of a theorem of Helton and Spitkovsky, generalizing their result in
the process.
