Editing Group theory (section)

===Algebraic geometry===
{{Main|Algebraic geometry}}
[[Algebraic geometry]] likewise uses group theory in many ways. [[Abelian variety|Abelian varieties]] have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. (For example the [[Hodge conjecture]] (in certain cases).) The one-dimensional case, namely [[elliptic curve]]s is studied in particular detail. They are both theoretically and practically intriguing.<ref>See the [[Birch and Swinnerton-Dyer conjecture]], one of the [[millennium problem]]s</ref> In another direction, [[toric variety|toric varieties]] are [[algebraic variety|algebraic varieties]] acted on by a [[torus]]. Toroidal embeddings have recently led to advances in [[algebraic geometry]], in particular [[resolution of singularities]].<ref>{{Citation | last1=Abramovich | first1=Dan | last2=Karu | first2=Kalle | last3=Matsuki | first3=Kenji | last4=Wlodarczyk | first4=Jaroslaw | title=Torification and factorization of birational maps | mr=1896232  | year=2002 | journal=[[Journal of the American Mathematical Society]] | volume=15 | issue=3 | pages=531–572 | doi=10.1090/S0894-0347-02-00396-X| arxiv=math/9904135 | s2cid=18211120 }}</ref>