In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to...

In this paper we use intersection theory to develop methods for obtaining lower bounds on the parameters of algebraic geometric error-correcting codes constructed from varieties of arbitrary dimension. The methods are sufficiently general to encompass many of the codes previously constructed from higher-dimensional varieties, as well as those coming from curves. And still, the bounds obtained are usually as good as the ones previously known (at least of the same order of magnitude with respect to the size of the ground field). Several examples coming from Deligne-Lusztig varieties, complete intersections of Hermitian hyper-surfaces, and from ruled surfaces (or more generally, projective bundles over a curve) are given.

Abstract. Let X ⊂ P n be a possibly singular hypersurface of degree d ≤ n, defined over a finite field Fq. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its ℓ-adic cohomology H i ( ¯ X, Qℓ), for ℓ ̸ = char(Fq), are divisible by q, without using the result on the existence of rational points by Ax and Katz [18]. 1.

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Abstract. Let X be a projective variety, homogeneous under a linear algebraic group. We show that the diagonal of X belongs to a unique irreducible component HX of the Hilbert scheme of X ×X. Moreover, HX is isomorphic to the “wonderful completion ” of the connected automorphism group of X; in particular, HX is nonsingular. We describe explicitly the degenerations of the diagonal in X × X, that is, the points of HX; these subschemes of X ×X are reduced and Cohen-Macaulay.

. We factor the virtual Poincar'e polynomial of every homogeneous space G=H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t 2u (t 2 \Gamma 1) r QG=H (t 2 ) for a polynomial QG=H with non-negative integer coefficients. Moreover, we show that QG=H (t 2 ) divides the virtual Poincar'e polynomial of every regular embedding of G=H, if H is connected.

Let X be a projective variety of dimension n defined over an algebraically closed field k. For X irreducible and non-singular, Matsusaka [Ma] constructed an abelian variety Alb (X) and a morphism α: X → Alb (X) (called the Albanese variety and mapping respectively), depending on the choice of a base-point on X, which is universal among the morphisms to abelian varieties (see Lang [La], Serre [Se] for other constructions). Over the field of complex numbers the existence of Alb (X) and α was known before, and has a purely Hodge-theoretic description (see Igusa [I] for the Hodge theoretic construction). Incidentally, the terminology “Albanese variety ” was introduced by A. Weil, for reasons explained in his commentary on the article [1950a] of Volume I of his collected works (see [W]), one of which is that the paper [Alb] of Albanese defines it (for a surface) as a quotient of the group of 0-cycles of degree 0 modulo an equivalence relation. Let CH n (X)deg 0 denote the Chow group of 0-cycles of degree 0 on X modulo rational equivalence. When X is irreducible and non-singular, a remarkable feature of the Albanese morphism α is that it factors through a regular homomorphism

We study from a geographical point of view fibrations of threefolds over smooth curves f: T − → B such that the general fibre is of general type. We prove the non-negativity of certain relative invariants under general hypotheses and give lower bounds for K3 T/B depending on other relative invariants. We also study the influence of the relative irregularity q(T)−g(B) on these bounds. A more detailed study of the lowest cases of the bounds is given.

Abstract. The space of smooth rational cubic curves in projective space P r (r ≥ 3) is a smooth quasi-projective variety, which gives us an open subset of the corresponding Hilbert scheme, the moduli space of stable maps, or the moduli space of stable sheaves. By taking its closure, we obtain three compactifications H, M, and S respectively. In this paper, we compare these compactifications. First, we prove that H is the blow-up of S along a smooth subvariety parameterizing planar stable sheaves. Next we prove that S is obtained from M by three blow-ups followed by three blow-downs and the centers are described explicitly. Using this, we calculate the cohomology of S.