  Let $\mathcal{X}$ be a projective, irreducible, nonsingular algebraic curve
over the finite field $\mathbb{F}_q$ with $q$ elements and let
$|\mathcal{X}(\mathbb{F}_q)|$ and $g(\mathcal X)$ be its number of rational
points and genus respectively. The Ihara constant $A(q)$ has been intensively
studied during the last decades, and it is defined as the limit superior of
$|\mathcal{X}(\mathbb{F}_q)|/g(\mathcal X)$ as the genus of $\mathcal X$ goes
to infinity. In 2012 Homma defined an analogue $D(q)$ of $A(q)$, where the
nonsingularity of $\mathcal X$ is dropped and $g(\mathcal X)$ is replaced with
the degree of $\mathcal X$. We will call $D(q)$ Homma's constant. In this
paper, upper and lower bounds for the value of $D(q)$ are found.
