  In this paper for each $n\ge g\ge 0$ we consider the moduli stack
$\widetilde{\mathcal U}^{ns}_{g,n}$ of curves
$(C,p_1,\ldots,p_n,v_1,\ldots,v_n)$ of arithmetic genus $g$ with $n$ smooth
marked points $p_i$ and nonzero tangent vectors $v_i$ at them, such that the
divisor $p_1+\ldots+p_n$ is nonspecial (has no $h^1$) and ample. With some mild
restrictions on the characteristic we show that it is a scheme, affine over the
Grassmannian $G(n-g,n)$. We also construct an isomorphism of
$\widetilde{\mathcal U}^{ns}_{g,n}$ with a certain relative moduli of
$A_\infty$-structures (up to an equivalence) over a family of graded
associative algebras parametrized by $G(n-g,n)$.
