## EQUISINGULAR DEFORMATIONS OF PLANE CURVES IN ARBITRARY CHARACTERISTIC (2006)

### BibTeX

@MISC{Campillo06equisingulardeformations,

author = {Antonio Campillo and Gert-martin Greuel and Christoph Lossen},

title = {EQUISINGULAR DEFORMATIONS OF PLANE CURVES IN ARBITRARY CHARACTERISTIC},

year = {2006}

}

### OpenURL

### Abstract

Dedicated to Joseph Steenbrink on the occasion of his sixtieth birthday Abstract. In this paper we develop the theory of equisingular deformations of plane curve singularities in arbitrary characteristic. We study equisingular deformations of the parametrization and of the equation and show that the base space of its semiuniveral deformation is smooth in both cases. Our approach through deformations of the parametrization is elementary and we show that equisingular deformations of the parametrization form a linear subfunctor of all deformations of the parametrization. This gives additional information, even in characteristic zero, the case which was treated by J. Wahl. The methods and proofs extend easily to good characteristic, that is, when the characteristic does not divide the multiplicity of any branch of the singularity. In bad characteristic, however, new phenomena occur and we are naturally led to consider weakly trivial respectively weakly equisingular deformations, that is, those which become trivial respectively equisingular after a finite and dominant base change. The semiuniversal base space for weakly equisingular

### Citations

167 |
A Singular Introduction to Commutative Algebra
- Greuel, Pfister
- 2002
(Show Context)
Citation Context ... �� �� �� EQUISINGULAR DEFORMATIONS IN ARBITRARY CHARACTERISTIC 33 Here we use the well known fact that if a map ϕ : A → B in AK is injective then the map Spec(ϕ) : Spec(B) → Spec(A) is dominant (cf. =-=[GP]-=-, Proposition A.3.8). If ϕ is finite then Spec(ϕ) is closed and hence Spec(ϕ) is surjective. (2) We comment the situation for (weakly) trivial deformations: If char(K) = 0 then it is known that if a d... |

165 | Functors of Artin rings
- Schlessinger
- 1968
(Show Context)
Citation Context ...bject of Def es es es induces a versal object of Def R←P R . Since Def R←P is unobstructed by Theorem 3.1, the same follows for Def es es R . Since Def R←P and Def R satisfy Schlessinger’s conditions =-=[Sch]-=- for the existence of a formal versal deformation (the first by Theorem 3.1, for the second this is well-known), the same holds for Def es es R . Now, it follows from [Fl, (5.2) Satz] that Def R has a... |

40 |
Singularities of Plane Curves
- Casas–Alvero
- 2000
(Show Context)
Citation Context ... From the equality JR = t d C and the fact that dimK R/C = 2δ, we get dimK R /( JR ) = δ + |d| , dimK m/(m · RJ) = δ + r − 1 + |d| . The fact that dimK(J/mJ) = 2 in characteristic 0 follows e.g. from =-=[Ca2]-=-. Proposition 5.5. The exact sequence (5.1) induces an exact sequence of R-modules where M sec R 0 −→ M sec R 1,es 1,es 1,es −→ T −→ T −→ T R/R R←R R −→ 0 , 1,es = T = 0 if the characteristic of K is ... |

39 |
Algebroid Curves in Positive Characteristic
- Campillo
- 1980
(Show Context)
Citation Context ...der) of ϕi and the r–tupel ord(ϕ) = (ordϕ1, . . .,ordϕr) the multiplicity (or order) of the parametrization ϕ. Note that ordϕi is the maximal integer mi s.t. ϕi(mP) ⊂ 〈ti〉 mi . Moreover, we have (cf. =-=[Ca]-=-) mt(f) = ordϕ1 + . . . + ordϕr. Definition 1.1. A deformation with sections of the parametrization of R over A ∈ AK is a commutative diagram with Cartesian squares ϕ R P � � K □ □ RA PA � A ϕA σ σ={σ... |

30 |
Equisingular deformations of plane algebroid curves, Trans
- Wahl
(Show Context)
Citation Context ...gebraic) base space of the semiuniversal deformation of R. The theory of equisingular deformations of plane curve singularities in characteristic zero has been initiated by J. Wahl in his thesis (cf. =-=[Wa]-=-). Wahl’s approach is different from ours as he considers only deformations of the equation and defines equisingularity by requiring equimultiplicity of the equation of the reduced total transform alo... |

23 | The hunting of invariants in the geometry of discriminants - Teissier - 1977 |

20 |
Families of varieties with prescribed singularities
- Greuel, Karras
- 1989
(Show Context)
Citation Context ...known that if a deformation η ∈ Def R (C) becomes trivial after some base change ϕ : C → C ′ then ϕ factors as C π ։ C tr → C ′ where C tr is a unique factor algebra of C such that πη is trivial (cf. =-=[GKa]-=-, Lemma 1.4). Hence Spec(C tr ) ⊂ Spec(C) is the unique maximal substratum over which η is trivial (and a family is weakly trivial iff it is trivial). The proof uses Schlessinger’s theory of functors ... |

18 | Deformations of plane curves with nodes and cusps - Wahl - 1974 |

17 | Simple singularities in positive characteristic - Greuel, Kröning - 1990 |

15 |
Polynomial ideals defined by infinitely near base points
- Zariski
- 1938
(Show Context)
Citation Context .... The set of essential points for R is denoted by Ess(R). The set Ess(R) will be considered for an embedded (good) resolution of R. By the theorem of resolution of singularities (cf. e.g. [Ca], [Li], =-=[Za]-=-) Ess(R) is finite. Definition 2.4. We define the multiplicity (or order) of a deformation with sections of the parametrization (ϕA, σ, σ), to be the r-tuple ord(ϕAσ, σ) := m = (m1, . . . , mr) such t... |

13 |
Introduction to Singularities and Deformations
- Greuel, Lossen, et al.
- 2007
(Show Context)
Citation Context ...f P on R. Although equimultiplicity of the parametrization is usually stronger than equimultiplicity of the equation, one can prove that Wahl’s functor ES and our functor Def es R are isomorphic (cf. =-=[GLS]-=-). Thus, we get, in characteristic zero, a new proof of Wahl’s result that the equisingularity stratum (which coincides then with the µ–constant stratum, where µ is the Milnor number) in the base spac... |

12 |
On complete ideals in regular local rings, In: Algebraic Geometry and Commutative Algebra in Honor of M
- Lipman
- 1987
(Show Context)
Citation Context ...smooth. The set of essential points for R is denoted by Ess(R). The set Ess(R) will be considered for an embedded (good) resolution of R. By the theorem of resolution of singularities (cf. e.g. [Ca], =-=[Li]-=-, [Za]) Ess(R) is finite. Definition 2.4. We define the multiplicity (or order) of a deformation with sections of the parametrization (ϕA, σ, σ), to be the r-tuple ord(ϕAσ, σ) := m = (m1, . . . , mr) ... |

8 | Ein Kriterium für die Offenheit der Versalität - Flenner - 1981 |

7 |
Contributions à la théorie de singularités
- Buchweitz
- 1981
(Show Context)
Citation Context ...i K[[ti]] ⊂ Quot(R) = i=1 r⊕ K((ti)). Note that, in good characteristic, we have di = mi − 1. Altogether, we get an exact sequence of R-modules, which are finite dimensional K-vector spaces (see also =-=[Bu]-=- and [GLS]). i=1 0 −→ MR −→ T 1 1 −→ T R/R R←R −→ T 1 R −→ R / RJ −→ 0 . All maps are obvious, except for the map T 1 R←R → T 1 R , which takes the class of . Note that RJ is an ideal of R as J is con... |

6 | On the equivalence of singularities - Hironaka - 1965 |

6 | A theorem on the monodromy of isolated singularities, Singularités à Cargèse - Lazzeri |

2 | Hamburger-Noether expansions over rings - Campillo - 1983 |

2 |
Intersection sur les surfaces reguliers
- Deligne
- 1973
(Show Context)
Citation Context ...0 R ⊂ T 0 R is identified with the K-vector space MR := T 0/( 0 R T R ∩ T 0 R ) . R/R Note that, in characteristic zero, we have MR = 0 as every derivation in T 0 R can be extended to one in T 0 (see =-=[De]-=-). However, if char(K) = p > 0, this is not true. REQUISINGULAR DEFORMATIONS IN ARBITRARY CHARACTERISTIC 27 For instance, if p ∤ q, the derivative ∂ ∂y is tangent to the curve {yp + xq = 0} and the i... |

1 |
G.-M.; Lossen, C.: Equisingular calculations for plane curve singularities
- Campillo, Greuel
(Show Context)
Citation Context ...ion of a semiuniversal equisingular deformation in good characteristic (that is, of the µ–constant stratum in characteristic 0). This has been implemented in the computer algebra system Singular (cf. =-=[CGL]-=- for a description of the algorithm). In bad characteristic, however, new phenomena occur. There are deformations which are not equisingular but become equisingular after some finite (and dominant) ba... |

1 | Commutative Ring Theory, Cambridge studies in advances mathematics 8 - Matsumara - 1986 |