  We establish a close connection between stable commutator length in free
groups and the geometry of sails (roughly, the boundary of the convex hull of
the set of integer lattice points) in integral polyhedral cones. This
connection allows us to show that the scl norm is piecewise rational linear in
free products of Abelian groups, and that it can be computed via integer
programming. Furthermore, we show that the scl spectrum of nonabelian free
groups contains elements congruent to every rational number modulo
$\mathbb{Z}$, and contains well-ordered sequences of values with ordinal type
$\omega^\omega$. Finally, we study families of elements $w(p)$ in free groups
obtained by surgery on a fixed element $w$ in a free product of Abelian groups
of higher rank, and show that $\scl(w(p)) \to \scl(w)$ as $p \to \infty$.
