The singular locus of the secant varieties of a smooth projective curve

Let X be a smooth irreducible non-degenerated projective curve in some projective space P^N. Let r be a positive integer such that 2r + 1 < N and let S_r(X) be the r-th secant variety of X. It is a variety of dimension 2r + 1. In this paper we prove that the singular locus is the (r - 1)-th secant variety S_{r- 1}(X) if X does not have any (2r + 2)-secant 2r-space divisor. Received: 26 November 2002     

Preservation of the injectivity of the Gauss map for general projections

Let X be a smooth irreducible quasi-projective variety of dimension n in P ^N with N ≥ 2n + 2. Let γ be its Gauss map, let be the embedding obtained from the general projection in P ^N and let γ′ be its Gauss map. We say that the general projection preserves the injectivity of the Gauss map if γ(Q) ≠ γ(Q′) implies γ′(Q) ≠ γ′ (Q′). We prove that this property holds in the following cases: N≥ 3n + 1; N ≥ 3n with n ≥ 2; N ≥ 3n−1 with n ≥ 4 and X does not contain a linear (n−1)-space. In case N = 3n−1 and X does contain a linear (n−1)-space (such smooth varieties exist) then the general projection does not preserver the injectivity of the Gauss map. This shows that there does not exist a straightforward kind of Bertini theorem for properties related to the Gauss map. The author is affiliated with the University at Leuven as a research fellow. This paper belongs to the FWO-project G.0318.06.     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Versality of linearly normal non-special embeddings

LetX⊂P ⁿ be a smooth non-degenerate non special linearly normal projective curve. Here we classify all such embeddings ofX such that for every hyperplaneM ofP ⁿ the family of all hyperplane sections ofX is a versal deformation of the zerodimensional schemeX∩M.

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Sunto SiaX una curva liscia e proiettiva. Qui si classificano le immersioni non-speciali linearmente normali diX inP ⁿ tali che per ogni iperpianoM diP ⁿ la famiglia delle sezioni iperpiane diX induce una deformazione versale diX∩M.

     

Curves in the double plane

We study in detail locally Cohen-Macaulay curves in P⁴ which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes H _d,g(2H) of lo-cally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that H _d,g(2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equiva-lence classes.     

On a result of Miyanishi-Masuda

Let X be an affine surface admitting a unique affine ruling and a -action. Assume that the ruling has a unique degenerate fibre and that this fibre is irreducible. In this paper we give a short proof of the following result of Miyanishi and Masuda: the universal covering of X is a hypersurface in the affine 3-space given by the equation x^my = z^d − 1, where m > 1. Received: 13 June 2005     

Rational Lagrangian fibrations on punctual Hilbert schemes of K3 surfaces

A rational Lagrangian fibration f on an irreducible symplectic variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov–Beauville square 0. This conjecture is proved in the case when V is the Hilbert scheme of d points on a generic K3 surface S of genus g under the hypothesis that its degree 2g−2 is a square times 2d−2. The construction of f uses a twisted Fourier–Mukai transform which induces a birational isomorphism of V with a certain moduli space of twisted sheaves on another K3 surface M, obtained from S as its Fourier–Mukai partner.     

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”.     

Barsotti–Tate groups and p-adic representations of the fundamental group scheme

On a scheme S over a base scheme B we study the category of locally constant BT groups, i.e. groups over S that are twists, in the flat topology, of BT groups defined over B. These groups generalize p-adic local systems and can be interpreted as integral p-adic representations of the fundamental group scheme of S/B (classifying finite flat torsors on the base scheme) when such a group exists. We generalize to these coefficients the Katz correspondence for p-adic local systems and show that they are closely related to the maximal nilpotent quotient of the fundamental group scheme.     

Cohomology of toric bundles

Let p: E B be a principal bundle with fibre and structure group the torus T ( ^*)ⁿ over a topological space B. Let X be a nonsingular projective T-toric variety. One has the X-bundle : E(X) B where E(X) = E × _T X, ([e,x]) = p(e). This is a Zariski locally trivial fibre bundle in case p: E B is algebraic. The purpose of this note is to describe (i) the singular cohomology ring of E(X) as an H ^* (B;)-algebra, (ii) the topological K-ring of K ^* (E(X)) as a K ^* (B)-algebra when B is compact. When p : E B is algebraic over an irreducible, nonsingular, noetherian scheme over , we describe (iii) the Chow ring of A ^* (E(X)) as an A ^* (B)-algebra, and (iv) the Grothendieck ring $\mathcal K$⁰ (E (X)) of algebraic vector bundles on E (X) as a $\mathcal K$⁰(B)-algebra.     

Some remarks on factoriality of certain hypersurfaces in \mathbb{P}^4

Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of ₂ ordinary double points, ₃ ordinary triple points and if ₂ + 4₃ < (k – 1)², then any smooth surface contained in V is a complete intersection on V.Received: 7 January 2004     

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

Rigidity and moduli space in Real Algebraic Geometry

Let R be a real closed field and let X be an affine algebraic variety over R. We say that X is universally map rigid (UMR for short) if, for each irreducible affine algebraic variety Z over R, the set of nonconstant rational maps from Z to X is finite. A bijective map from an affine algebraic variety over R to X is called weak change of the algebraic structure of X if it is regular and φ⁻¹ is a Nash map, which preserves nonsingular points. We prove the following rigidity theorem: every affine algebraic variety over R is UMR up to a weak change of its algebraic structure. Let us introduce another notion. Let Y be an affine algebraic variety over R. We say that X and Y are algebraically unfriendly if all the rational maps from X to Y and from Y to X are trivial, i.e., Zariski locally constant. From the preceding theorem, we infer that, if dim (X)≥1, then there exists a set of weak changes of the algebraic structure of X such that, for each t,s ∈ R with t≠s, and are algebraically unfriendly. This result implies the following expected fact: For each (nonsingular) affine algebraic variety X over R of positive dimension, the natural Nash structure of X does not determine the algebraic structure of X. In fact, the moduli space of birationally nonisomorphic (nonsingular) affine algebraic varieties over R, which are Nash isomorphic to X, has the same cardinality of R. This result was already known under the special assumption that R is the field of real numbers and X is compact and nonsingular. The author is a member of GNSAGA of CNR, partially supported by MURST and European Research Training Network RAAG 2002–2006 (HPRN–CT–00271).     

The singularH 2,2-invariant quartic surfaces in ℙ3

AnH _2,2-invariant quartic surface in ₃ is a quartic surface in ₃ invariant under the Heisenberg groupH _2,2 of level (2, 2), the family ofH _2,2-invariant quartic surfaces is parametrized by ₄. For each ₄, the corresponding quartic surfaceX will be a Kummer surface, ifX is singular. The equation for { = 0} ₄ parametrizing all Kummer surfaces is well known. We find another more symmetric form (with respect to a 5-dimensional representation of the symmetric group S₆) for this equation.The aim of this note is to describe all singularH _2,2-invariant quartic surfaces in ₃.     

Generation of k-jets on toric varieties

The notion of a k-convex -support function for a toric variety is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the -support function . Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k. Received October 22, 1997; in final form January 12, 1998     

Complex surfaces polarized by an ample and spanned line bundle of genus three

Let X be a complex connected projective nonsingular algebraic surface endowed with an ample line bundle L, which is spanned by its global sections. Pairs (X, L) as above, with sectional genus g(X, L)=1+(L·(K _X L))/2=3 are classified by means of the main techniques of adjunction theory.     

The <Emphasis Type="Italic">L</Emphasis>-series of a cubic fourfold

We study the L-series of cubic fourfolds. Our main result is that, if X/C is a special cubic fourfold associated to some polarized K3 surface S, defined over a number field K and satisfying , then X has a model over K such that the L-series of the primitive cohomology of X/K can be expressed in terms of the L-series of S/K. This allows us to compute the L-series for a discrete dense subset of cubic fourfolds in the moduli spaces of certain special cubic fourfolds. We also discuss a concrete example.     

Bipolarized threefolds with hyperelliptic curve sections

A classification of smooth complex projective threefoldsX polarized by two very ample line bundlesL andM is given, under the assumption that two general elements of |L| and |M| intersect transversally along a smooth hyperelliptic curve.

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Sunto Si fornisce una classificazione delle terne (X,L,M), doveX è una varietà algebrica proiettiva complessa liscia di dimensione 3 edL,M sono due fibrati lineari molto ampi suX, tali che l’intersezione di due elementi generici di |L| ed |M| sia una curva iperellittica.

     

Three dimensional divisorial extremal neighborhoods

In this paper we study divisorial extremal neighborhoods such that 0 ∈ X is a cA_n type threefold terminal singularity, and Γ=f(E) is a smooth curve, where E is the f-exceptional divisor. We view a divisorial extremal neighborhood as a one parameter smoothing of certain surface singularities, and based on this we give a classification of such neighborhoods.