107 research outputs found
Containment results for ideals of various configurations of points in P^N
No full textGuided by evidence coming from a few key examples and attempting to unify previous work of Chudnovsky, Esnault-Viehweg, Eisenbud-Mazur, Ein-Lazarsfeld-Smith, Hochster-Huneke and Bocci-Harbourne, Harbourne and Huneke recently formulated a series of conjectures that relate symbolic and regular powers of ideals of fat points in PN. In this paper we propose another conjecture along the same lines, and we verify it and the conjectures of Harbourne and Huneke for a variety of configurations of points. The research that led to the present paper was partially supported by a grant of the group GNSAGA of INdAM
Comparing powers and symbolic powers of ideals
No full textWe develop tools to study the problem of containment of symbolic powers I^(m) in powers I^r for a homogeneous ideal I in a polynomial ring k[PN] in N + 1 variables over an arbitrary algebraically closed field k. We obtain results on the structure of the set of pairs (r, m) such that I^(m) is contained in I^r. As corollaries, we show that I^2 contains I^(3) whenever S is a finite generic set of points in P^2 (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith [Invent. Math. 144 (2001), pp. 241-252] and Hochster-Huneke [Invent. Math. 147 (2002), pp. 349-369] are optimal for every fixed dimension and codimension
The resurgence of ideals of points and the containment problem
No full textWe relate properties of linear systems on X to the question of when I^r contains I^(m) in the case that I is the homogeneous ideal of a finite set of distinct points p_1,...,p_n in P^2, where X is the surface obtained by blowing up the points. We obtain complete answers for when I^r contains I^(m) when the points p_i's lie on a smooth conic or when the points are general and n ≤ 9
ON plane rational curves and the splitting of the tangent bundle
No full textGiven a rational plane curve, we give new approaches to determining the splitting of the pullback of the cotangent bundle. We give applications to determining the minimal resolution for ideals of fat points in the plane
Stable Postulation and Stable Ideal Generation: Conjectures for Fat Points in the Plane
No full textIt is an open problem to determine the Hilbert function and graded Betti
numbers for the ideal of a fat point subscheme supported at general points
of the projective plane. In fact, there is not yet even a general explicit conjecture
for the graded Betti numbers. Here we formulate explicit asymptotic
conjectures for both problems. We work over an algebraically closed field K
of arbitrary characteristic
Projectively Normal but Superabundant Embeddings of Rational Surfaces in Projective Space
No full textPeuliar embeddings of certain rational surfaces are studied which are projectively normal, but the liner system adopted is superabundan
Expecting the unexpected: Quantifying the persistence of unexpected hypersurfaces
No full textThe independence of imposed vanishing conditions is a foundational issue for a wide range of research in algebraic geometry. In this spirit, if X⊂Pn is a reduced subscheme, we say that X admits an unexpected hypersurface of degree t and multiplicity m if requiring multiplicity m at a general point P fails to impose the expected number of conditions on the linear system of hypersurfaces of degree t containing X. Conditions which either guarantee the occurrence of unexpected hypersurfaces, or which ensure that they cannot occur, are not well understood. Research to date has made surprising connections to root systems, hyperplane arrangements, and generic splitting types of vector bundles, among other diverse topics. In this paper we introduce new methods for studying unexpectedness, such as the use of generic initial ideals and partial elimination ideals to clarify when it can and when it cannot occur. We also exhibit algebraic and geometric properties of X which in some cases guarantee and in other cases preclude X having certain kinds of unexpectedness. In addition, we formulate a new way of quantifying unexpectedness (our AV sequence), which allows us to detect the extent to which unexpectedness persists as t increases but t−m remains constant. Finally, we study to what extent we can detect unexpectedness from the Hilbert function of X
Star Configurations in P^n
No full textStar configurations are certain unions of linear subspaces of
projective space. They have appeared in several different contexts: the study of extremal Hilbert functions for fat point schemes in the plane; the study of secant varieties of some classical algebraic varieties; the study of the resurgence of projective schemes. In this paper we study some algebraic properties of the ideals defining star configurations, including getting partial results about Hilbert functions, generators and minimal free resolutions of the ideals and their symbolic powers. We also show that their symbolic powers define arithmetically Cohen–Macaulay subschemes and we obtain
results about the primary decompositions of the powers of the
ideals. As an application, we compute the resurgence for the ideal of the codimension n −1 star configuration in Pn in the monomial case (i.e., when the number of hyperplanes
is n +1)
On the discriminant of spanned line bundles
No full textLet X be an irreducible smooth complex projective variety of dimension n, L a nontrivial spanned line bundle on X, V \subseteq H^0(X,L) a vector subspace od global sections spanning L with dim(V)=N+1.
The discriminant variety D(X,V) parameterizes the singular elements of |V|. By Bertini's theorem one can write dim(D(X,V))=N-1-k with k a non-negative integer. An upper bound on k in terms of N, n, and the dimension of the fibers of the map defined by V is established. Moreover, triplets (X,L,V) for which k is near the maximum are classified
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