On Zariski's theorem in positive characteristic

Tyomkin, Ilya. "On Zariski's theorem in positive characteristic." Journal of the European Mathematical Society 015.5 (2013): 1783-1803. <http://eudml.org/doc/277410>.

@article{Tyomkin2013,
abstract = {In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $-K_S. C+p_g(C)-1$, where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $\texttt \{dim\} (V)=-K_S.C+p_g(C)-1$ does not imply the nodality of $C$ even if $C$ belongs to the smooth locus of $S$, and construct reducible Severi varieties on weighted projective planes in positive characteristic, parameterizing irreducible reduced curves of given geometric genus in a given very ample linear system.},
author = {Tyomkin, Ilya},
journal = {Journal of the European Mathematical Society},
keywords = {curves on algebraic surfaces; Severi varieties; curves on algebraic surfaces; Severi varieties},
language = {eng},
number = {5},
pages = {1783-1803},
publisher = {European Mathematical Society Publishing House},
title = {On Zariski's theorem in positive characteristic},
url = {http://eudml.org/doc/277410},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Tyomkin, Ilya
TI - On Zariski's theorem in positive characteristic
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 5
SP - 1783
EP - 1803
AB - In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $-K_S. C+p_g(C)-1$, where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $\texttt {dim} (V)=-K_S.C+p_g(C)-1$ does not imply the nodality of $C$ even if $C$ belongs to the smooth locus of $S$, and construct reducible Severi varieties on weighted projective planes in positive characteristic, parameterizing irreducible reduced curves of given geometric genus in a given very ample linear system.
LA - eng
KW - curves on algebraic surfaces; Severi varieties; curves on algebraic surfaces; Severi varieties
UR - http://eudml.org/doc/277410
ER -