Details
| Original language | English |
|---|---|
| Pages (from-to) | 1-45 |
| Number of pages | 45 |
| Journal | Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques |
| Volume | 137 |
| Issue number | 1 |
| Early online date | 1 Dec 2022 |
| Publication status | Published - Jun 2023 |
Abstract
In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski’s theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.
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In: Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques, Vol. 137, No. 1, 06.2023, p. 1-45.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - On the Severi problem in arbitrary characteristic
AU - Christ, Karl
AU - He, Xiang
AU - Tyomkin, Ilya
N1 - Funding Information: IT is partially supported by the Israel Science Foundation (grant No. 821/16). KC is supported by the Israel Science Foundation (grant No. 821/16) and by the Center for Advanced Studies at BGU. XH was supported by the ERC Consolidator Grant 770922 – BirNonArchGeom.
PY - 2023/6
Y1 - 2023/6
N2 - In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski’s theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.
AB - In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi’s original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski’s theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.
UR - http://www.scopus.com/inward/record.url?scp=85143266941&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2005.04134
DO - 10.48550/arXiv.2005.04134
M3 - Article
AN - SCOPUS:85143266941
VL - 137
SP - 1
EP - 45
JO - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques
JF - Publications Mathematiques de l'Institut des Hautes Etudes Scientifiques
SN - 0073-8301
IS - 1
ER -