Theorem 2.33 of Moduli of curves by Harris and Morrison.

The idea is to start with a curve $C$ of genus at least two and consider the family of degree 3 branched coverings ramified at a unique point. This gives a one-dimensional complete family of curves of a certain genus. Iterating this construction, one gets complete families of arbitrary dimension.

Theorem 2.33 of Moduli of curves by Harris and Morrison gives a construction with weaker bounds.

The idea is to start with a curve $C$ of genus at least two and consider the family of degree 3 branched coverings ramified at a unique point. This gives a one-dimensional complete family of curves of a certain genus. Iterating this construction, one gets complete families of arbitrary dimension.

Theorem 2.33 of Moduli of curves by Harris and Morrison.

The idea is to start with a curve $C$ of genus at least two and consider the family of degree 3 branched coverings ratified at a unique point. This gives a one-dimensional complete family of curves of a certain genus. Iterating this construction, one gets complete families of arbitrary dimension.

Theorem 2.33 of Moduli of curves by Harris and Morrison.

The idea is to start with a curve $C$ of genus at least two and consider the family of degree 3 branched coverings ramified at a unique point. This gives a one-dimensional complete family of curves of a certain genus. Iterating this construction, one gets complete families of arbitrary dimension.