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How about $$x_1^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-1} \quad ?$$$$x_1^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-2} \quad ?$$
This is a fairly well-known example which seems to satisfy your criteria.
How about $$x_1^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-1} \quad ?$$
This is a fairly well-known example which seems to satisfy your criteria.
How about $$x_1^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-2} \quad ?$$
This is a fairly well-known example which seems to satisfy your criteria.
How about $$x_0^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-1} \quad ?$$$$x_1^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-1} \quad ?$$
This is a fairly well-known example which seems to satisfy your criteria.
How about $$x_0^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-1} \quad ?$$
This is a fairly well-known example which seems to satisfy your criteria.
How about $$x_1^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-1} \quad ?$$
This is a fairly well-known example which seems to satisfy your criteria.
How about $$x_0^{q-1} + \cdots + x_{q-1}^{q-1} = 0 \subset \mathbb{P}^{q-1} \quad ?$$
This is a fairly well-known example which seems to satisfy your criteria.
