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The codegree density (F) of an r-graph F is the largest number such that there are F-free r-graphs G on n vertices such that every set of r - 1 vertices is contained in at least (\gamma - o(1))n edges. For the Fano plane PG(2, 2), Mubayi showed that  \gamma(PG(2, 2)) = 1/2. For general projective geometries we obtain the upper bound \gamma(PG(m, q)) 1/m. We show that equality holds whenever m = 2 and q is odd, and whenever m = 3 and q is 2 or 3. We also give examples of r-graphs with codegree densities equal to 1-1/k for all r and k.
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