TY - JOUR

T1 - On stabilizers of algebraic function fields of one variable

AU - Geyer, Wulf Dieter

AU - Jarden, Moshe

AU - Razon, Aharon

N1 - Publisher Copyright:
© 2017 by Walter de Gruyter Berlin/Boston.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - Let K be a fixed algebraic closure of an infinite field K. We consider an absolutely integral curve Γ in PKn PKn with n ≥ 2. The curve ΓKΓK should have only finitely many inflection points, finitely many double tangents, and there exists no point in PKnPKn through which infinitely many tangents to ΓKΓK go. In addition there exists a prime number q such that ΓKΓK has a cusp of multiplicity q and the multiplicities of all other points of ΓKΓK are at most q. Under these assumptions, we construct a non-empty Zariski-open subset O of PKnPKn such that if n ≥ 3, the projection from each point o O(K) birationally maps Γ onto an absolutely integral curve Γ in PKn-1 PKn-1 with the same properties as Γ (keeping q unchanged). If n = 2, then the projection from each o O(K) maps Γ onto PK1PK1 and leads to a stabilizing element t of the function field F of Γ over K. The latter means that F/K(t) is a finite separable extension whose Galois closure FF is regular over K.

AB - Let K be a fixed algebraic closure of an infinite field K. We consider an absolutely integral curve Γ in PKn PKn with n ≥ 2. The curve ΓKΓK should have only finitely many inflection points, finitely many double tangents, and there exists no point in PKnPKn through which infinitely many tangents to ΓKΓK go. In addition there exists a prime number q such that ΓKΓK has a cusp of multiplicity q and the multiplicities of all other points of ΓKΓK are at most q. Under these assumptions, we construct a non-empty Zariski-open subset O of PKnPKn such that if n ≥ 3, the projection from each point o O(K) birationally maps Γ onto an absolutely integral curve Γ in PKn-1 PKn-1 with the same properties as Γ (keeping q unchanged). If n = 2, then the projection from each o O(K) maps Γ onto PK1PK1 and leads to a stabilizing element t of the function field F of Γ over K. The latter means that F/K(t) is a finite separable extension whose Galois closure FF is regular over K.

KW - Function feld

KW - absolutely integral curve

UR - http://www.scopus.com/inward/record.url?scp=85018780117&partnerID=8YFLogxK

U2 - 10.1515/advgeom-2016-0026

DO - 10.1515/advgeom-2016-0026

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AN - SCOPUS:85018780117

SN - 1615-715X

VL - 17

SP - 131

EP - 174

JO - Advances in Geometry

JF - Advances in Geometry

IS - 2

ER -