Geometry of obstructed families of curves

We study certain equisingular families of curves with quasihomogeneous singularity of minimal obstructness (i.e. h¹=1). We show that our families always have expected codimension. Moreover they are either non-reduced with smooth reduction or decompose into two smooth components of expected codimension that intersect non-transversally or are reduced irreducible non-smooth varieties which have smooth singular locus with sectional singularity of type A₁. On the other hand there is an example of an equisingular family of curves with multiple quasihomogeneous singularities of minimal obstructness which is smooth but has wrong codimension. We use algorithms of computer algebra as a technical tool.     

Topological equisingularity of hypersurfaces with 1-dimensional critical set

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 Lê numbers then it is equisingular at the critical set, and hence topologically equisingular (answering a question of D. Massey [13]). It is worth to remark that this does not happen for higher dimensional critical set [5]. We use these topological triviality results to modify the definition of singularity stem present in the literature, introducing and characterising topological stems (being this concept closely related with Arnold?s series of singularities). We provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly. Finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non-flat deformation with singular special fibre and smooth generic fibre.     

Some remarks on the study of good contractions

LetX be a complex projective variety with log terminal singularities admitting an extremal contraction in terms of Minimal Model Theory, i.e. a projective morphism φ:X→Z onto a normal varietyZ with connected fibers which is given by a (high multiple of a) divisor of the typeK _x+rL, wherer is a positive rational number andL is an ample Cartier divisor. We first prove that the dimension of anu fiberF of φ is bigger or equal to (r-1) and, if φ is birational, thatdimF≥r, with the equalities if and only ifF is the projective space andL the hyperplane bundle (this is a sort of “relative” version of a theorem of Kobayashi-Ochiai). Then we describe the structure of the morphism φ itself in the case in which all fibers have minimal dimension with the respect tor. If φ is a birational divisorial contraction andX has terminal singularities we prove that φ is actually a “blow-up”.     

On normal surface singularities and a problem of enriques

We construct families of normal surface singularities with the following property: given any fiat projective connected family V →B of smooth, irreducible, minimal algebraic surfaces, the general singularity in one of our families cannot occur, analytically, on any algebraic surfaces which is Irrationally equivalent to a surface in V→B. In particular this holds for V→B consisting of a single rational surface, thus answering negatively to a long standing problem posed by F. Enriques. In order to prove the above mentioned results, wo develop a general, though elementary, method, based on the consideration of suitable correspondences, for comparing a given family of minimal surfaces with a family of surface singularities. Specifically the method in question gives us the possibility of comparing the parameters on which the two families depend, thus leading to the aforementioned results.     

Remarks on families of singular curves with hyperelliptic normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus p_g ? 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result that relies on deformation theory and bending-and-breaking of rational curves in Sym²(S). We also give examples of families of such curves.     

The multiplicity of abelian covers of splice quotient singularities

From a resolution graph with certain conditions, Neumann and Wahl constructed an equisingular family of surface singularities called splice quotients. For this class some fundamental analytic invariants have been computed from their resolution graph. In this paper we give a method to compute the multiplicity of an abelian covering of a splice quotient from its resolution graph and the Galois group.     

n-Dimensional Fano varieties of wild representation type

The aim of this work is to contribute to the classification of projective varieties according to their representation type, providing examples of n  -dimensional varieties of wild representation type, for arbitrary n?2$n?2$. More precisely, we prove that all Fano blow-ups of Pⁿ${\mathbb{P}}^n$ at a finite number of points are of wild representation type exhibiting families of dimension of order r²$r^2$ of simple (hence, indecomposable) ACM rank r   vector bundles for any r?n$r?n$. In the two dimensional case, the vector bundles that we construct are moreover Ulrich bundles and μ-stable with respect to certain ample divisor.     

Foliations with persistent singularities

Let ω be a differential q-form defining a foliation of codimension q in a projective variety. In this article we study the singular locus of ω in various settings. We relate a certain type of singularities, which we name persistent, with the unfoldings of ω, generalizing previous work done on foliations of codimension 1 in projective space. We also relate the absence of persistent singularities with the existence of a connection in the sheaf of 1-forms defining the foliation.     

Fundamental groups of open K3 surfaces, Enriques surfaces and Fano 3-folds

We investigate when the fundamental group of the smooth part of a K3 surface or Enriques surface with Du Val singularities, is finite. As a corollary we give an effective upper bound for the order of the fundamental group of the smooth part of a certain Fano 3-fold. This result supports Conjecture A below, while Conjecture A (or alternatively the rational-connectedness conjecture in Kollar et al. (J. Algebra Geom. 1 (1992) 429) which is still open when the dimension is at least 4) would imply that every log terminal Fano variety has a finite fundamental group.     

Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian

We introduce various families of irreducible homaloidal hypersurfaces in projective space Pr, for all r?3. Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of hypersurfaces that are dual to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. The latter family fit non-classical versions of de Jonquières transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan-Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision into the present knowledge.     

Tilting on non-commutative rational projective curves

In this article we introduce a new class of non-commutative projective curves and show that in certain cases the derived category of coherent sheaves on them has a tilting complex. In particular, we prove that the right bounded derived category of coherent sheaves on a reduced rational projective curve with only nodes and cusps as singularities, can be fully faithfully embedded into the right bounded derived category of the finite dimensional representations of a certain finite dimensional algebra of global dimension two. As an application of our approach we show that the dimension of the bounded derived category of coherent sheaves on a rational projective curve with only nodal or cuspidal singularities is at most two. In the case of the Kodaira cycles of projective lines, the corresponding tilted algebras belong to a well-known class of gentle algebras. We work out in details the tilting equivalence in the case of the Weierstrass nodal curve zy ² = x ³ + x ² z.     

Exponential sums and singular hypersurfaces

We give upper bounds for the absolute value of exponential sums in several variables attached to certain polynomials with coefficients in a finite field. This bounds are given in terms of invariants of the singularities of the projective hypersurface defined by its highest degree form. For exponential sums attached to the reduction modulo a power of a large prime of a polynomial f with integer coefficients and veryfying a certain condition on the singularities of its highest degree form, we give a bound in terms of the dimension of the Jacobian quotient . Received: 3 November 1997     

On the divisor class group of double solids

For a double solid V→ℙ_3> branched over a surface B⊂ℙ₃(ℂ) with only ordinary nodes as singularities, we give a set of generators of the divisor class group in terms of contact surfaces of B with only superisolated singularities in the nodes of B. As an application we give a condition when H ^* (˜V , ℤ) has no 2-torsion. All possible cases are listed if B is a quartic. Furthermore we give a new lower bound for the dimension of the code of B. Received: 16 November 1998     

Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers

An SI-sequence is a finite sequence of positive integers which is symmetric, unimodal and satisfies a certain growth condition. These are known to correspond precisely to the possible Hilbert functions of graded Artinian Gorenstein algebras with the weak Lefschetz property, a property shared by a nonempty open set of the family of all graded Artinian Gorenstein algebras having a fixed Hilbert function that is an SI sequence. Starting with an arbitrary SI-sequence, we construct a reduced, arithmetically Gorenstein configuration G of linear varieties of arbitrary dimension whose Artinian reduction has the given SI-sequence as Hilbert function and has the weak Lefschetz property. Furthermore, we show that G has maximal graded Betti numbers among all arithmetically Gorenstein subschemes of projective space whose Artinian reduction has the weak Lefschetz property and the given Hilbert function. As an application we show that over a field of characteristic zero every set of simplicial polytopes with fixed h-vector contains a polytope with maximal graded Betti numbers.     

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to minus n+2 hyperplanes.Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a real n-torus as its fiber. The base accommodates the geometric genus of a hypersurface V. Its homotopy type is a wedge of h^n,o(V) spheres Sⁿ.     

Non degenerate projective curves with very degenerate hyperplane section

In this paper, we study the Hilbert scheme of non degenerate locally Cohen- Macaulay projective curves with general hyperplane section spanning a linear space of dimension 2 and minimal Hilbert function. The main result is that those curves are almost always the general element of a generically smooth component H_n,d,g of the corresponding Hilbert scheme. Moreover, we show that the curves with maximal cohomology almost always correspond to smooth points of H_n,d,g.All the authors were partially supported by Acción Integrada Italia-España, HI2000-0091, and by the Italian counterpart of the project.     

Global holomorphic 2-forms and pluricanonical systems on threefolds

The change of zero locus of a global holomorphic 2-form on a threefold under birational transformations is investigated. It is proved that existence of 2-forms with certain conditions on their zero loci on a threefold of nonnegative Kodaira dimension limits types of terminal singularities appearing on its minimal models. As a result of the restriction on the types of terminal singularities and Reid's Riemann-Roch formula, a universal bound N is found such that the linear system NK defines a birational map from a threefold of general type admitting those 2-forms, where K is the canonical bundle of the threefold. Received March 10, 2000 / Published online October 11, 2000     

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class.In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian–Yang–Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with coefficients in R is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.     

Rigidity for orientable functors

In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K-theory of algebraically closed fields. Besides K-theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes.