More refined information is available, for larger values of "g", but in these cases canonical curves are not generally complete intersections, and the description requires more consideration of commutative algebra. The field started with Max Noether's theorem: the dimension of the space of quadrics passing through "C" as embedded as canonical curve is ("g" − 2)("g" − 3)/2. Petri's theorem, often cited under this name and published in 1923 by Karl Petri (1881–1955), states that for "g" at least 4 the homogeneous ideal defining the canonical curve is generated by its elements of degree 2, except for the cases of (a) trigonal curves and (b) non-singular plane quintics when "g" = 6. In the exceptional cases, the ideal is generated by the elements of degrees 2 and 3. Historically speaking, this result was largely known before Petri, and has been called the theorem of Babbage-Chisini-Enriques (for Dennis Babbage who completed the proof, Oscar Chisini and Federigo Enriques). The terminology is confused, since the result is also called the Noether–Enriques theorem. Outside the hyperelliptic cases, Noether proved that (in modern language) the canonical bundle is normally generated: the symmetric powers of the space of sections of the canonical bundle map onto the sections of its tensor powers. This implies for instance the generation of the quadratic differentials on such curves by the differentials of the first kind; and this has consequences for the local Torelli theorem. Petri's work actually provided explicit quadratic and cubic generators of the ideal, showing that apart from the exceptions the cubics could be expressed in terms of the quadratics. In the exceptional cases the intersection of the quadrics through the canonical curve is respectively a ruled surface and a Veronese surface.