  An integer matrix $\mathbf{A}$ is $\Delta$-modular if the determinant of each
$\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix of
$\mathbf{A}$ has absolute value at most $\Delta$. The study of $\Delta$-modular
matrices appears in the theory of integer programming, where an open conjecture
is whether integer programs defined by $\Delta$-modular constraint matrices can
be solved in polynomial time if $\Delta$ is considered constant. The conjecture
is only known to hold true when $\Delta \in \{1,2\}$. In light of this
conjecture, a natural question is to understand structural properties of
$\Delta$-modular matrices. We consider the column number question -- how many
nonzero, pairwise non-parallel columns can a rank-$r$ $\Delta$-modular matrix
have? We prove that for each positive integer $\Delta$ and sufficiently large
integer $r$, every rank-$r$ $\Delta$-modular matrix has at most $\binom{r+1}{2}
+ 80\Delta^7 \cdot r$ nonzero, pairwise non-parallel columns, which is tight up
to the term $80\Delta^7$. This is the first upper bound of the form
$\binom{r+1}{2} + f(\Delta)\cdot r$ with $f$ a polynomial function. Underlying
our results is a partial list of matrices that cannot exist in a
$\Delta$-modular matrix. We believe this partial list may be of independent
interest in future studies of $\Delta$-modular matrices.
