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{{short description|Regular 3-D curves are shape and size determined by their curvature and torsion}} ...Torsion of curves|torsion]].<ref>{{citation|title=Differential Geometry of Curves and Surfaces|first1=Thomas F.|last1=Banchoff|first2=Stephen T.|last2=Lovett ...

In [[mathematics]], there are several theorems of [[Helmut Hasse]] that are sometimes called '''Hasse's theorem''': * [[Hasse's theorem on elliptic curves]] ...

'''Gromov's compactness theorem''' can refer to either of two mathematical theorems: ...(topology)]] on the existence of limits of sequences of pseudoholomorphic curves ...

...al theorem of curves]] describes classification of [[Regular curve|regular curves]] in space up to [[Translation (geometry)|translation]] and [[rotation]]. ==Fundamental theorems of mathematical topics== ...

: Consider two non-singular curves ''C'' and ''{{prime|C}}'' having the same [[genus (mathematics)|genus]] ''g * {{cite book | author=Coolidge, J. L. | title=A Treatise on Algebraic Plane Curves | location=New York | publisher=Dover | page=135 | year=1959 | isbn=0-486-6 ...

{{short description|On limiting subsequences of sequence of pseudoholomorphic curves with uniform energy bound}} ...rve. This theorem, and its generalizations to punctured pseudoholomorphic curves, underlies the compactness results for flow lines in [[Floer homology]] and ...

...2</math> are ''k''-isomorphic for ''k'' any [[perfect field]], so are the curves.<ref>James S. Milne, ''Jacobian Varieties'', Corollary 12.2 in {{Harvtxt|Co ...s). Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the [[local Torelli theorem]]. Secondly, to othe ...

{{Algebraic curves navbox}} [[Category:Algebraic curves]] ...

{{short description|Describes rational torsion points on elliptic curves over the integers}} ...ational number|rational]] [[Torsion (algebra)|torsion]] points on elliptic curves over the integers. It is named for [[Trygve Nagell]] and [[Élisabeth Lutz]] ...

{{short description|One of several theorems in different areas of mathematics}} ...ematician)|John M. Sullivan]] has observed that Schur's theorem applies to curves of finite total curvature (the statement is slightly different). ...

{{About|the concept in geometry|the concept in [[mathematical optimization]]|Fenche ...2 \pi/L</math>, where <math>L</math> is the length of the curve. The only curves of this type whose total absolute curvature equals <math>2\pi</math> and wh ...

{{Short description|Three-dimensional smooth curves with small total curvature must be unknotted}} {{about|curvature of knots|the theorem concerning straight-line embeddings of plana ...

...ed the way for the [[Grothendieck–Hirzebruch–Riemann–Roch theorem]] proved about three years later. ...''Noether's formula'''). Weil's Riemann–Roch theorem for vector bundles on curves, and the Riemann–Roch theorem for algebraic surfaces (see below), are inclu ...

...generated by [[Dehn twist]]s along 3''g'' &minus; 1 specific simple closed curves in the surface, where ''g'' denotes the [[genus (mathematics)|genus]] of th [[Category:Theorems in topology]] ...

...pages=|oclc=405546184}}</ref>^{pg 207}. The main structure theorem about this group is the [[Mordell–Weil theorem]] which shows this group is in fac ...last=Booher|first=Jeremy|date=|title=The Mordell–Weil theorem for elliptic curves|url=https://www.math.arizona.edu/~jeremybooher/expos/mordellweil.pdf|url-st ...

==Main theorems of Brill–Noether theory== ...pecial divisors. One goal of the theory is to 'count constants', for those curves: to predict the dimension of the space of special divisors (up to [[linear ...

==Differential geometry of curves and surfaces== ===[[Differential geometry of curves]]=== ...

'''[[Helmut Hasse|Hasse]]'s theorem on elliptic curves''', also referred to as the Hasse bound, provides an estimate of the number ...ization of the Hasse bound to higher [[Geometric genus|genus]] [[algebraic curves]] is the Hasse–Weil bound. This provides a bound on the number of points o ...

{{short description|Statement about cubic curves in the projective plane}} ...''Cayley–Bacharach theorem''' is a statement about [[cubic curve]]s (plane curves of degree three) in the [[projective plane]] {{math|'''P'''²}}. ...

...''C'' on ''V''. It says, roughly speaking, that the space spanned by such curves (up to [[linear equivalence]]) has a one-dimensional subspace on which it i ...t is known to be a [[finitely-generated abelian group]], and the result is about its [[tensor product]] with the rational number field. Therefore, ρ(''V'') ...