Geometry of the tetrahedron space

Let X^° be the space of all labeled tetrahedra in P³. In [E. Babson, P.E. Gunnells, R. Scott, A smooth space of tetrahedra, Adv. Math. 165(2) (2002) 285-312] we constructed a smooth symmetric compactification of X^°. In this article we show that the complement is a divisor with normal crossings, and we compute the cohomology ring .     

The doubles of a numerical semigroup

Let S be a numerical semigroup and let p be a positive integer. Then the quotient is also a numerical semigroup. When p=2 we say that is half of the numerical semigroup S. Dually, we say that S is a double of the numerical semigroup . We characterize the set of all doubles of a numerical semigroup. We also give some alternative proofs and improvements for some results that we find in previous papers.     

Separators of points on algebraic surfaces

For a finite set of points X⊆Pⁿ and for a given point P∈X, the notion of a separator of P in X (a hypersurface containing all the points in X except P) and of the degree of P in X, (the minimum degree of these separators) has been largely studied. In this paper we extend these notions to a set of points X on a projectively normal surface S⊆Pⁿ, considering as separators arithmetically Cohen-Macaulay curves and generalizing the case S=P² in a natural way. We denote the minimum degree of such curves as and we study its relation to . We prove that if S is a variety of minimal degree these two terms are explicitly related by a formula, whereas only an inequality holds for other kinds of surfaces.     

Resolutions of Segre embeddings of projective spaces of any dimension

This paper deals with syzygies of the ideals of Segre embeddings. Let d≥3 and . We prove that satisfies Green-Lazarsfeld’s property N_p if and only if p≤3.     

On varieties of almost minimal degree III: Tangent spaces and embedding scrolls

Let X⊂P^r be a variety of almost minimal degree which is the projected image of a rational normal scroll from a point p outside of . In this paper we study the tangent spaces at singular points of X and the geometry of the embedding scrolls of X, i.e. the rational normal scrolls Y⊂P^r which contain X as a codimension one subvariety.     

Families of low dimensional determinantal schemes

A scheme X⊂Pⁿ of codimension c is called standard determinantal if its homogeneous saturated ideal can be generated by the t×t minors of a homogeneous t×(t+c−1) matrix (f_ij). Given integers a₀≤a₁≤?≤a_t+c−2 and b₁≤?≤b_t, we denote by the stratum of standard determinantal schemes where f_ij are homogeneous polynomials of degrees a_j−b_i and is the Hilbert scheme (if n−c>0, resp. the postulation Hilbert scheme if n−c=0).Focusing mainly on zero and one dimensional determinantal schemes we determine the codimension of in and we show that is generically smooth along under certain conditions. For zero dimensional schemes (only) we find a counterexample to the conjectured value of appearing in Kleppe and Miró-Roig (2005) [25].     

Independence of ? in Lafforgue's theorem

Let X be a smooth curve over a finite field of characteristic p, let ?≠p be a prime number, and let be an irreducible lisse -sheaf on X whose determinant is of finite order. By a theorem of L. Lafforgue, for each prime number ?′≠p, there exists an irreducible lisse -sheaf on X which is compatible with , in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for and are equal. We prove an “independence of ?” assertion on the fields of definition of these irreducible ?′-adic sheaves : namely, that there exists a number field F such that for any prime number ?′≠p, the -sheaf above is defined over the completion of F at one of its ?′-adic places.     

Geometric degree of finite extensions of mappings from a smooth variety

Let f:V→W be a finite polynomial mapping of algebraic subsets V,W of and , respectively, with n≤m. It is known that f can be extended to a finite polynomial mapping . Moreover, it is known that, if V,W are smooth of dimension k,4k+2≤n=m, and f is dominated on every component (without vertical components) then there exists a finite polynomial extension such that , where means the number of points in the generic fiber of h. In this note we improve this result. Namely we show that there exists a finite polynomial extension such that .     

Computation of blowing up centers

Given a birational projective morphism of quasi-projective varieties . We want to find the ideal sheaf over X such that the blowing up of X along corresponds to f. In this paper we approach the problem from two directions, solving two subcases. First we present a method that determines when f is the composition of blowing ups along known centers, then by another method, we compute directly from .     

The compactified Picard scheme of the compactified Jacobian

Let C be an integral projective curve in any characteristic. Given an invertible sheaf L on C of degree 1, form the corresponding Abel map , which maps C into its compactified Jacobian, and form its pullback map , which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, double points, then is known to be an isomorphism. We prove that always extends to a map between the natural compactifications, , and that the extended map is an isomorphism if C has, at worst, ordinary nodes and cusps.     

Equivariant-constructible Koszul duality for dual toric varieties

For affine toric varieties X and defined by dual cones, we define an equivalence of categories between mixed versions of the equivariant derived category and the derived category of sheaves on which are locally constant with unipotent monodromy on each orbit. This equivalence satisfies the Koszul duality formalism of Beilinson, Ginzburg, and Soergel.     

Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire

We determine explicitly the irreducible components of the singular locus of any Schubert variety for being an algebraically closed field of arbitrary characteristic. We also describe the generic singularities along each of them.The case of covexillary Schubert varieties was solved in an earlier work of the author [Ann. Inst. Fourier 51 (2) (2001) 375]. Here, we first exhibit some irreducible components of the singular locus of X_w, by describing the generic singularity along each of them. Let Σ_w be the union of these components. As mentioned above, the equality is known for covexillary varieties, and we base our proof of the general case on this result. More precisely, we study the exceptional locus of certain quasi-resolutions of a non-covexillary Schubert variety X_w, and we relate the intersection of these loci to Σ_w. Then, by induction on the dimension, we can establish the equality.