  In the present paper, we study the relationship between deformation
quantizations and Frobenius-projective structures defined on an algebraic curve
in positive characteristic. A Frobenius-projective structure is an analogue of
a complex projective structure on a Riemann surface, which was introduced by Y.
Hoshi. Such an additional structure has some equivalent objects, e.g., a
dormant $\mathrm{PGL}_2$-oper and a projective connection having a full set of
solutions. The main result of the present paper provides a canonical
construction of a Frobenius-constant quantization on the cotangent space minus
the zero section on an algebraic curve by means of a Frobenius-projective
structure. It may be thought of as a positive characteristic analogue of a
result by D. Ben-Zvi and I. Biswas. Finally, we give a higher-dimensional
variant of this result, as proved by I. Biswas in the complex case.
