Editing Algebraic geometry (section)

===19th and early 20th century===
It took the simultaneous 19th century developments of [[non-Euclidean geometry]] and [[Abelian integral]]s in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by [[Edmond Laguerre]] and [[Arthur Cayley]], who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea of ''homogeneous polynomial forms'', and more specifically [[quadratic form]]s, on projective space. Subsequently, [[Felix Klein]] studied projective geometry (along with other types of geometry) from the viewpoint that the geometry on a space is encoded in a certain class of [[transformation group|transformations]] on the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather than the projective linear transformations which were normally regarded as giving the fundamental [[Kleinian geometry]] on projective space, they concerned themselves also with the higher degree [[birational transformation]]s. This weaker notion of congruence would later lead members of the 20th century [[Italian school of algebraic geometry]] to classify [[algebraic surface]]s up to [[birational isomorphism]].

The second early 19th century development, that of Abelian integrals, would lead [[Bernhard Riemann]] to the development of [[Riemann surface]]s.

In the same period began the algebraization of the algebraic geometry through [[commutative algebra]]. The prominent results in this direction are [[Hilbert's basis theorem]] and [[Hilbert's Nullstellensatz]], which are the basis of the connection between algebraic geometry and commutative algebra, and [[Francis Sowerby Macaulay|Macaulay]]'s [[multivariate resultant]], which is the basis of [[elimination theory]]. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed by [[singularity theory]] and computational algebraic geometry.{{efn|A witness of this oblivion is the fact that [[Van der Waerden]] removed the chapter on elimination theory from the third edition (and all the subsequent ones) of his treatise ''Moderne algebra'' (in German).{{citation needed|date=January 2020}}}}