Automorphism Groups of Algebraic Curves G´abor Korchm´aros Universit`adegli Studi della Basilicata Italy Joint work with M. Giulietti Combinatorics and Computer Algebra, CoCoA 2015, July 19-25 2015 Forth Collins G´abor Korchm´aros Automorphism Groups of Algebraic Curves Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; Some background from Algebraic Geometry G´abor Korchm´aros Automorphism Groups of Algebraic Curves K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); G´abor Korchm´aros Automorphism Groups of Algebraic Curves Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; G´abor Korchm´aros Automorphism Groups of Algebraic Curves C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: G´abor Korchm´aros Automorphism Groups of Algebraic Curves C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); G´abor Korchm´aros Automorphism Groups of Algebraic Curves 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map G´abor Korchm´aros Automorphism Groups of Algebraic Curves Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C G´abor Korchm´aros Automorphism Groups of Algebraic Curves Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, G´abor Korchm´aros Automorphism Groups of Algebraic Curves Two algebraic curves with the same plane model are birationally isomorphic; Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); G´abor Korchm´aros Automorphism Groups of Algebraic Curves Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; G´abor Korchm´aros Automorphism Groups of Algebraic Curves Some background from Algebraic Geometry Algebraic curve:= projective (absolutely irreducible, non-singular) algebraic variety of dimension 1 in a projective space PG(r; K); K:= field of characteristic p ≥ 0, Assumption (w.l.g.): K is algebraically closed field; Every algebraic curve X has a plane model C: C := irreducible (in general singular) curve in PG(2; K); C is a model of X if 9 a map 8 PG(r; ) ! PG(2; ) <> K K Φ = (Y0; Y1;:::; Yr ) ! (X0; X1; X2) :>X!C Xi = Ui (Y0; Y1;:::; Yr ) homogeneous polynomials of degree d ≥ 1, Φ is injective and surjective (apart from a finite number of points); Two algebraic curves with the same plane model are birationally isomorphic; G´abor Korchm´aros Automorphism Groups of Algebraic Curves Let C be a plane model of X ; C:= (projective, irreducible) plane curve with homogeneous equation F (X0; X1; X2) = 0; homogeneous equation ! affine equation; F (X0; X1; X2) ! f (X ; Y ) = F (1; X ; Y ); α:=automorphism of C (and of X ); 8 PG(2; ) ! PG(2; ) <> K K 0 0 0 α = (X0; X1; X2) ! (X0; X1; X2) :>C!C 0 Xi = Ui (X0; X1; X2) homogeneous polynomials of degree d ≥ 1, α is surjective and injective on C (apart from a finite number of points); Aut(X ):=group of all automorphisms of X ; G < Aut(X ):= tame when p - jGj, otherwise non-tame. Concept of automorphism G´abor Korchm´aros Automorphism Groups of Algebraic Curves C:= (projective, irreducible) plane curve with homogeneous equation F (X0; X1; X2) = 0; homogeneous equation ! affine equation; F (X0; X1; X2) ! f (X ; Y ) = F (1; X ; Y ); α:=automorphism of C (and of X ); 8 PG(2; ) ! PG(2; ) <> K K 0 0 0 α = (X0; X1; X2) ! (X0; X1; X2) :>C!C 0 Xi = Ui (X0; X1; X2) homogeneous polynomials of degree d ≥ 1, α is surjective and injective on C (apart from a finite number of points); Aut(X ):=group of all automorphisms of X ; G < Aut(X ):= tame when p - jGj, otherwise non-tame.