In math.AG/0108089 we gave sufficient conditions for the irreducibility of the family V of irreducible curves in the linear system |D| with precisely r singular points of topological respectively analytical types S1,...,Sr on several classes of smooth projective surfaces S. The conditions where of the form (tau'(S1)+2)^2 +...+ (tau'(Sr)+2)^2 < c*(D-K)^2, where tau' is some invariant of singularity types, K is the canonical divisor of S and c is some constant. In the present paper we improve this condition, that is the constant c, by a factor 9.
In 1985 Joe Harris proved the long standing claim of Severi that equisingular families of nodal plane curves are irreducible whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor D on a smooth projective surface S it thus makes sense to look for conditions which ensure that the family V_D(S_1,...,S_r) of irreducible curves in the linear system |D| with precisely r singular points of types S_1,...,S_r is irreducible. Considering different surfaces including general surfaces in P^3 and products of curves, we produce a sufficient con…
We explore some equisingularity criteria in one parameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski's discriminant criterion. We prove that topological triviality implies smoothness of the normalized surface. Examples are given to show that Witney regularity and equisaturation are not stable under the blow-up of the singular locus nor under the Nash modification.
We prove the analogue of Viehweg's hyperbolicity conjecture for Whitney equisingular families of projective varieties with Gorenstein rational singularities whose geometric generic fiber has a good minimal model. Namely, for such families with maximal variation, the base spaces are of log general type. The main new ingredient is the use of intersection complexes as Hodge modules in the construction of logarithmic Higgs sheaves by Viehweg-Zuo and Popa-Schnell. This construction suggests an equisingular stratification of the moduli space of varieties of general type, with each stratum being hype…
We study geometric properties of certain obstructed equisingular families of projective hypersurfaces with emphasis on smoothness, reducibility, being reduced, and having expected dimension. In the case of minimal obstructness, we give a detailed description of such families corresponding to quasihomogeneous singularities. Next we study the behavior of these properties with respect to stable equivalence of singularities. We show that under certain conditions, stabilization of singularities ensures the existence of a reduced component of expected dimension. For minimally obstructed families the…
Very few examples of obstructed equsingular families of curves on surfaces other than the projective plane are known. Combining results from Westenberger and Hirano with an idea from math.AG/9802009 we give in the present paper series of examples of families of irreducible curves with simple singularities on surfaces in projective three-space which are not T--smooth, i.e. do not have the expected dimension, and we compare this with conditions (showing the same asymptotics) which ensure the existence of a T--smooth component.
We investigate the following question: let $C$ be an integral curve contained in a smooth complex algebraic surface $X$; is it possible to deform $C$ in $X$ into a nodal curve while preserving its geometric genus? We affirmatively answer it in most cases when $X$ is a Del Pezzo or Hirzebruch surface, and in some cases when $X$ is a $K3$ surface. Partial results are given for all surfaces with numerically trivial canonical class. We also give various examples for which the answer is negative.
We present new results on equisingularity and equinormalizability of families with isolated non-normal singularities (INNS) of arbitrary dimension. We define a $δ$-invariant and a $μ$-invariant for an INNS and prove necessary and sufficient numerical conditions for equinormalizability and weak equinormalizability using $δ$ and $μ$. Moreover, we determine the number of connected components of the Milnor fibre of an arbitrary INNS. For families of generically reduced curves, we investigate the topological behavior of the Milnor fibre and characterize topological triviality of such families. Fina…
We investigate the equisingularity question for $1$-parameter deformation families of mixed polynomial functions $f_t(\mathbf{z},\bar{\mathbf{z}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the corresponding family of mixed hypersurfaces $f_t^{-1}(0)$ is Whitney equisingular (and hence topologically equisingular) and satisfies the Thom condition.
We study the behavior of limits of tangents in topologically equivalent spaces. In the context of families of generically reduced curves, we introduce the $s$-invariant of a curve and we show that in a Whitney equisingular family with the property that the $s$-invariant is constant along the parameter space, the number of tangents of each curve of the family is constant. In the context of families of isolated surface singularities, we show through examples that Whitney equisingularity is not sufficient to ensure that the tangent cones of the family are homeomorphic. We explain how the existenc…
In this survey, we report on progress concerning families of projective curves with fixed number and fixed (topological or analytic) types of singularities. We are, in particular, interested in numerical, universal and asymptotically proper sufficient conditions to guarantee the nonemptyness, T-smoothness and irreducibility of the variety of all projective curves with prescribed singularities in a fixed linear system. We also discuss the analogous problem for hypersurfaces of arbitrary dimension with isolated singularities, and we close with a section on open problems and conjectures.
We consider flat families of reduced curves on a smooth surface S such that each member C has the same number of singularities of fixed singularity types and the corresponding (locally closed) subscheme H of the Hilbert scheme of S. We are mainly concerned with analytic resp. topological singularity types and give a sufficient condition for the smoothness of H (at C). Our results for S=P^2 seem to be quite sharp for families of cuves of small degree d.
We focus on topological equisingularity of families of holomorphic function germs with 1-dimensional critical set. We introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular. We show that if a family of germs with 1-dimensional critical set has constant generic Le numbers then it is equisingular at the critical set, and hence topologically equisingular (answering a question of D. Massey). We use this to modify the definition of singularity stem present in the literature, introducing and charac…
In this paper we give some criteria for a family of generically reduced plane curve singularities to be equinormalizable. The first criterion is based on the $δ$-invariant of a (non-reduced) curve singularity which is introduced by Brücker-Greuel (\cite{BG}). The second criterion is based on the I-equisingularity of a $k$-parametric family ($k\geq 1$) of generically reduced plane curve singularities, which is introduced by Nobile (\cite{No}) for one-parametric families. The equivalence of some kinds of equisingularities of a family of generically reduced plane curve singularities is also studi…
In the 1970s O. Zariski introduced a general theory of equisingularity for algebroid and algebraic hypersurfaces over an algebraically closed field of characteristic zero. His theory builds up on understanding the dimensionality type of hypersurface singularities, notion defined recursively by considering the discriminants loci of successive "generic" corank 1 projections. The theory of singularities of dimensionality type 1, that is the ones appearing generically in codimension 1, was developed by Zariski in his foundational papers on equisingular families of plane curve singularities. In thi…
Francesco Severi showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor D on a smooth projective surface it thus makes sense to look for conditions which ensure that the family V of irreducible curves in the linear system |D| with precisely r singular points of types S1,...,Sr is T-smooth. Considering different surfaces including the projective plane, general surfaces in projective 3-space, products of curves and geometrical…
Greuel, Lossen and Shustin gave a general sufficient numerical condition for the T-smoothness (smoothness and expected dimension) of equisingular families of plane curves. This condition involves a new invariant γfor plane curve singularities, and it is conjectured to be asymptotically proper. In math.AG/0308247, similar sufficient numerical conditions are obtained for the T-smoothness of equisingular families on various classes surfaces. These conditions involve a series of invariants γ_a, 0 <= a <= 1, with γ_1=γ. In the present paper we compute (respectively give bounds for) these invariants…
The Complete Intersection Discrepancy of a Curve II: Families of Curves
Institute of Mathematics and Informatics Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Department of Mathematics Northeastern University（东北大学数学系）；Institut de Mathématiques de Jussieu-Paris Rive Gauche Centre national de la recherche scientifique (CNRS)（巴黎左岸苏邦数学研究所 法国国家科学研究中心）
We construct exponentially large collections of pairwise distinct equisingular deformation families of irreducible plane curves sharing the same sets of singularities. The fundamental groups of all curves constructed are abelian.
In math.AG/0108089, math.AG/0212090 and math.AG/0308247 we gave numerical conditions which ensure that an equisingular family is irreducible respectively T-smooth. Combining results by Greuel, Lossen and Shustin and an idea from math.AG/9802009 we give in the present paper series of examples of families of irreducible curves on surfaces in projective three-space with only ordinary multiple points which are reducible and where at least one component does not have the expected dimension. The examples show that for families of curves with ordinary multiple points the conditions for T-smoothness i…
Let $S$ be a smooth projective surface over $\mathbb{C}$. We study the local and global geometry of the nested Hilbert scheme of points $S^{[n,n+1,n+2]}$. In particular, we show that $S^{[n,n+1,n+2]}$ is an irreducible local complete intersection with klt singularities. In addition, we compute the Picard group of $S^{[n,n+1,n+2]}$ when $h^1(S,\mathcal{O}_S) = 0$. From the irreducibility of $S^{[n,n+1,n+2]}$, we deduce irreducibility for four other infinite families of nested Hilbert schemes. We give the first explicit example of a reducible nested Hilbert scheme, which allows us to show that $…
We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{P}^n$ i.e. the open subscheme parameterising zero-dimensional Gorenstein subschemes of $\mathbb{P}^n$ of degree $d$. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $d\leq 13$ and find its components when $d = 14$. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by--product, we give equations of the fourth secant variety to the $d$-th Veronese reembedding of $…
We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite number of given analytic or topological singularity types, there exist a plane (irreducible) curve of degree $d$ having singular points of the given type as its only singularities. The set of all such curves is a quasi-projective variety, which we call an equisingular family (ESF). We describe, in terms of numerical invariants of the curves and their singularit…
Let P^2_r be the projective plane blown up at r generic points. Denote by E_0,E_1,...,E_r the strict transform of a generic straight line on P^2 and the exceptional divisors of the blown-up points on P^2_r respectively. We consider the variety V of all irreducible curves C in |dE_0-d_1E_1-...-d_rE_r| with k nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptyness. Moreover, we extend our conditions for the smoothness and the irreducibility on families of reducible curves. For r<10 we give the complete answ…
We study families V of curves in P^2 of degree d having exactly r singular points of given topological or analytic types. We derive new sufficient conditions for V to be T-smooth (smooth of the expected dimension), respectively to be irreducible. For T-smoothness these conditions involve new invariants of curve singularities and are conjectured to be asymptotically proper, i.e., optimal up to a constant factor. To obtain the results, we study the Castelnuovo function, prove the irreducibility of the Hilbert scheme of zero-dimensional schemes associated to a cluster of infinitely near points of…
