Function Fields, Curves Introduction Function Fields vs. Curves - PowerPoint PPT Presentation

Algorithmics of Function Fields 1 Function Fields, Curves, Lecture 1 Global Sections Function Fields, Curves Introduction Function Fields vs. Curves and Global sections Function Fields Curves Representation and Definition Representation Via Affine Curve Summer School UNCG 2016 Completion Normalisation Magma Florian Hess Global Sections Outline Sheaves Diagonalisation Global Sections Grothendiecks Theorem Riemann-Roch Special Models Magma Exercises 1 / 44

Algorithmics of Function Fields 1 Function Fields, Curves, Global Sections Introduction Function Fields vs. Curves Function Fields Curves Introduction Representation and Definition Representation Via Affine Curve Completion Normalisation Magma Global Sections Outline Sheaves Diagonalisation Global Sections Grothendiecks Theorem Riemann-Roch Special Models Magma Exercises 2 / 44

Function Fields vs. Curves Algorithmics of Function Fields 1 Function Fields, Curves, Global Function fields vs. regular complete curves: Sections ◮ Essentially boil down to the same thing - there is an Introduction equivalence of categories. Function Fields vs. Curves ◮ If base field is C then there is another equivalence of Function Fields Curves categories, to compact Riemann surfaces and covering Representation maps. and Definition Representation ◮ So using one term over the other is more a socialogical Via Affine Curve Completion Normalisation question about one’s mathematical genesis or point of Magma view ... Global Sections ◮ Best to know all three ... Outline Sheaves Diagonalisation Global Sections Curves can also be singular, this gives some added ways of Grothendiecks Theorem Riemann-Roch expressing matters. Special Models Magma Exercises 3 / 44

Function Fields Algorithmics of Function Fields 1 Function Fields, Curves, Let K be a field. An algebraic function field of one variable is a Global Sections field extension F / K of transcendence degree one. Introduction This means that there is x ∈ F such that x is transcendental Function Fields vs. Curves over K and F / K ( x ) is finite. Function Fields Curves Representation The exact constant field of F / K is the algebraic closure K ′ of and Definition Representation K in F . Via Affine Curve Completion Normalisation Magma The extension F / K ′ is also an algebraic function field of one Global variable, the x from above is still transcendental over K ′ and Sections Outline F / K ′ ( x ) is finite. Sheaves Diagonalisation Global Sections Grothendiecks In theory one can always assume w.l.o.g. that K ′ = K . In Theorem Riemann-Roch practice one can not or should not. Special Models Magma Exercises 4 / 44

Separating Elements Algorithmics of Function Fields 1 Function Fields, Curves, The element x is called separating for F / K if F / K ( x ) is Global Sections separable. It is a theorem that if K is perfect then there is always a separating element for F / K . Introduction Function Fields vs. Curves Fields of characteristic zero, finite fields and algebraically closed Function Fields Curves fields are perfect. Any algebraic extension field of a perfect Representation field is perfect. and Definition Representation Via Affine Curve Example. The polynomial y 2 + x 2 + t ∈ F 2 ( t , x )[ y ] is Completion Normalisation Magma irreducible and purely inseparable. Thus Global Sections F = F 2 ( t , x )[ y ] / � y 2 + x 2 + t � Outline Sheaves Diagonalisation Global Sections is a purely inseparable field extension of degree two of F 2 ( t , x ). Grothendiecks Theorem Riemann-Roch Then F / F 2 ( t ) is an algebraic function field without a Special Models Magma separating element. Exercises 5 / 44

Local rings and Points Algorithmics of Function Fields 1 Function Fields, Curves, Global Sections We give a “function field” based approach to curves in the Introduction spirit of Hartshorne I.6, including singular curves. Function Fields vs. Curves Function Fields Curves Let F / K be an algebraic function field. A subring of F / K is a Representation proper subring O of F with K × ⊆ O × and Quot( O ) = F . and Definition Representation Via Affine Curve Completion If O is subring of F / K and a local ring with maximal ideal m Normalisation Magma we call it a point P of F / K with local ring O P = O and max- Global imal ideal m P = m . Sections Outline Sheaves Diagonalisation A place of F / K is regarded as point of F / K . Global Sections Grothendiecks Theorem Riemann-Roch Special Models Magma Exercises 6 / 44

Domination Algorithmics of Function Fields 1 Function Fields, Curves, Global Sections Introduction Let P und Q be points of F / K . We say that P is dominated Function Fields vs. Curves by Q if O P ⊆ O Q and m P ⊆ m Q holds. Function Fields Curves Representation We define supp( P ) as the set of places Q of F / K such that P and Definition Representation is dominated by Q . Via Affine Curve Completion Normalisation Magma Theorem. The sets supp( P ) are non-empty and finite. The Global residue class fields O P / m P are finite over K . Sections Outline Sheaves Diagonalisation Global Sections Grothendiecks Theorem Riemann-Roch Special Models Magma Exercises 7 / 44

Sets of Points and Curves Algorithmics of Function Fields 1 Function Fields, Curves, We will only consider sets U of points of F / K that are Global Sections ◮ admissible, i.e. almost all points of U are places. ◮ separated, i.e. for every place Q of F / K there is at most Introduction Function Fields one P ∈ U such that P is dominated by Q . vs. Curves Function Fields Curves Let U c denote the set of places of F / K that are not contained Representation and Definition in ∪ P ∈ U supp( P ). Then U is called cofinite, complete, and Representation affine if U c is finite, empty and non-empty respectively. Via Affine Curve Completion Normalisation Magma A curve C over K is an admissible separated cofinite set of Global Sections points of F / K . Outline Sheaves Diagonalisation The function field of C is K ( C ) = F . Global Sections Grothendiecks Theorem A point P ∈ C is regular if P is a place, otherwise singular. Riemann-Roch Special Models The curve is regular if all points of C are regular. Magma Exercises 8 / 44

Subrings Algorithmics of Function Fields 1 Function Let P ∈ C and U ⊆ C . We define O C , P = O P and Fields, Curves, Global Sections O C ( U ) = ∩ P ∈ U O C , P , Introduction where the empty intersection is defined as F . Function Fields vs. Curves Function Fields Suppose U is affine. Theorem. Curves Representation 1. The rings O C ( U ) are subrings of F / K and the maps and Definition Representation Via Affine Curve P �→ O C ( U ) ∩ m P and m �→ O C ( U ) m Completion Normalisation Magma give mutually inverse bijections from U to the set of Global non-zero maximal ideals of O C ( U ). Sections Outline 2. Every point in U is regular if and only if O C ( U ) is a Sheaves Diagonalisation Dedekind domain. Global Sections Grothendiecks Theorem 3. With D U ( f ) = { P ∈ U | f �∈ m P } for f ∈ O C ( U ), Riemann-Roch Special Models Magma O C ( D U ( f )) = O C ( U )[ f − 1 ] . Exercises 9 / 44

Affine Curves Algorithmics of Function Fields 1 Function Fields, Curves, Global Sections If R is a subring of F / K we define Specm( R ) to be the set of Introduction points of F / K defined by R m where m ranges over the maximal Function Fields vs. Curves ideals of R . Function Fields Curves Representation The map C �→ O C ( C ) gives an inclusion-reversing Theorem. and Definition Representation bijection of the set of affine curves C over K with K ( C ) = F Via Affine Curve Completion to the set of subrings R of F / K that are finitely generated Normalisation Magma K -algebras. Its inverse is given by R �→ Specm( R ). Global Sections Outline Sheaves This provides the link to the usual definition of affine curves. Diagonalisation Global Sections Grothendiecks Theorem Riemann-Roch Special Models Magma Exercises 10 / 44

Curves as Topological Spaces ∗ Algorithmics of Function Fields 1 Function Fields, Curves, Global Sections Let C be a curve over K . A subset U of C is called open if U Introduction is empty or C \ U is finite. Function Fields vs. Curves Function Fields Curves Representation Theorem. Let C be a curve over K . and Definition Representation 1. Then C with its open sets is a topological space. Via Affine Curve Completion 2 Moreover, it is an irreducible, one-dimensional T 1 -space Normalisation Magma and any open subset of C is quasicompact. Global Sections 3. If C is affine the sets D C ( f ) form a basis of the open sets Outline Sheaves of C . Diagonalisation Global Sections Grothendiecks Theorem Riemann-Roch Special Models Magma Exercises 11 / 44