# On Zariski's theorem in positive characteristic

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 5, page 1783-1803
- ISSN: 1435-9855

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topTyomkin, 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 -

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