Points In Projective Space Research Articles

Chudnovsky's conjecture and the stable Harbourne-Huneke containment for general points

In our previous work with Grifo and Hà, we showed the stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of sufficiently many general points in PN. In this paper, we establish the conjectures for all remaining cases, and hence, give the affirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecture for the general points in PN for all N. Our new technique is to develop the Cremona reduction process that provides effective lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective spaces.

Waldschmidt constants in projective spaces

In this note we introduce a Waldschmidt decomposition of divisors which might be viewed as a generalization of Zariski decomposition based on the effectivity rather than the nefness of divisors. As an immediate application we prove a recursive formula providing new effective lower bounds on Waldschmidt constants of very general points in projective spaces. We use these bounds in order to verify Demailly's conjecture in a number of new cases.

Determinantal Varieties From Point Configurations on Hypersurfaces

Abstract We consider the scheme $X_{r,d,n}$ parameterizing $n$ ordered points in projective space $\mathbb {P}^{r}$ that lie on a common hypersurface of degree $d$. We show that this scheme has a determinantal structure, and we prove that it is irreducible, Cohen–Macaulay, and normal. Moreover, we give an algebraic and geometric description of the singular locus of $X_{r,d,n}$ in terms of Castelnuovo–Mumford regularity and $d$-normality. This yields a characterization of the singular locus of $X_{2,d,n}$ and $X_{3,2,n}$.

Mini-Workshop: Subvarieties in Projective Spaces and Their Projections

The major goals of this workshop are to lay paths for a systematic study of geproci (and related, e.g., projecting to almost complete intersections or full intersections) sets of points in projective spaces, study algebraic properties of their ideals (e.g. in the spirit of the Cayley-Bacharach properties), and to identify the most promising new directions for study.

Duality for asymptotic invariants of graded families

The starting point of this paper is a duality for sequences of natural numbers which, under mild hypotheses, interchanges subadditive and superadditive sequences and inverts their asymptotic growth constants.We are motivated to explore this sequence duality since it arises naturally in at least two important algebraic-geometric contexts. The first context is Macaulay-Matlis duality, where the sequence of initial degrees of the family of symbolic powers of a radical ideal is dual to the sequence of Castelnuovo-Mumford regularity values of a quotient by ideals generated by powers of linear forms. This philosophy is drawn from an influential paper of Emsalem and Iarrobino. We generalize this duality to differentially closed graded filtrations of ideals.In a different direction, we establish a duality between the sequence of Castelnuovo-Mumford regularity values of the symbolic powers of certain ideals and a geometrically inspired sequence we term the jet separation sequence. We show that this duality underpins the reciprocity between two important geometric invariants: the multipoint Seshadri constant and the asymptotic regularity of a set of points in projective space.

The Hilbert function of general unions of lines, double lines and double points in projective spaces

The Hilbert function of general unions of lines, double lines and double points in projective spaces

A Cayley–Bacharach theorem and plane configurations

In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley–Bacharach condition. In particular, by bounding the number of points satisfying the Cayley–Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.

Chudnovsky’s conjecture and the stable Harbourne–Huneke containment

We investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky’s Conjecture and the stable version of the Harbourne–Huneke containment conjectures for a general set of sufficiently many points.

Demailly's Conjecture and the containment problem

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space . We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of this containment holds for generic determinantal ideals and defining ideals of star configurations.

Maximizing expected powers of the angle between pairs of points in projective space

Among probability measures on d-dimensional real projective space, one which maximizes the expected angle $$\arccos (\frac{x}{|x|}\cdot \frac{y}{|y|})$$ between independently drawn projective points x and y was conjectured to equidistribute its mass over the standard Euclidean basis $$\{e_0,e_1,\ldots , e_d\}$$ by Fejes Tóth (Acta Math Acad Sci Hung 10:13–19, 1959. https://doi.org/10.1007/BF02063286 ). If true, this conjecture evidently implies the same measure maximizes the expectation of $$\arccos ^\alpha (\frac{x}{|x|}\cdot \frac{y}{|y|})$$ for any exponent $$\alpha > 1$$ . The kernel $$\arccos ^\alpha (\frac{x}{|x|}\cdot \frac{y}{|y|})$$ represents the objective of an infinite-dimensional quadratic program. We verify discrete and continuous versions of this milder conjecture in a non-empty range $$\alpha > \alpha _{\Delta ^d} \ge 1$$ , and establish uniqueness of the resulting maximizer $${\hat{\mu }}$$ up to rotation. We show $${\hat{\mu }}$$ no longer maximizes when $$\alpha <\alpha _{\Delta ^d}$$ . At the endpoint $$\alpha =\alpha _{\Delta ^d}$$ of this range, we show another maximizer $$\mu $$ must also exist which is not a rotation of $${\hat{\mu }}$$ . For the continuous version of the conjecture, an “Appendix A” provided by Bilyk et al in response to an earlier draft of this work combines with the present improvements to yield $$\alpha _{\Delta ^d}<2$$ . The original conjecture $${\alpha _{\Delta ^d}}=1$$ remains open (unless $$d=1$$ ). However, in the maximum possible range $$\alpha >1$$ , we show $${\hat{\mu }}$$ and its rotations maximize the aforementioned expectation uniquely on a sufficiently small ball in the $$L^\infty $$ -Kantorovich–Rubinstein–Wasserstein metric $$d_\infty $$ from optimal transportation; the same is true for any measure $$\mu $$ which is mutually absolutely continuous with respect to $${\hat{\mu }}$$ , but the size of the ball depends on $$\alpha ,d$$ , and $$\Vert \frac{d {\hat{\mu }}}{d\mu }\Vert _{\infty }$$ .

A Cayley–Bacharach theorem for points in Pn

We prove a Cayley-Bacharach-type theorem for points in projective space $\mathbb{P}^n$ that lie on a complete intersection of $n$ hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections.

Degree bounds for local cohomology

Let R be a non-negatively graded Cohen-Macaulay ring with R_0 a Cohen-Macaulay factor ring of a local Gorenstein ring. Let d be the dimension of R, m be the maximal homogeneous ideal of R, and M be a finitely generated graded R-module. It has long been known how to read information about the socle degrees of the local cohomology module H_m^0(M) from the twists in position d in a resolution of M by free R-modules. It has also long been known how to use local cohomology to read valuable information from complexes which approximate resolutions in the sense that they have positive homology of small Krull dimension. The present paper reads information about the maximal generator degree (rather than the socle degree) of H_m^0M from the twists in position d-1 (rather than position d) in an approximate resolution of M. We apply the local cohomology results to draw conclusions about the maximum generator degree of the second symbolic power of the prime ideal defining a monomial curve and the second symbolic power of the ideal defining a finite set of points in projective space. There is an application to general hyperplane sections of subschemes of projective space over an infinite field. There is an application of the local cohomology techniques to partial Castelnuovo-Mumford regularity. An application to the ideals generated by the lower order Pfaffians of an alternating matrix will appear in a future paper. One additional application to the study of blow-up algebras appears in a separate paper.

Unexpected hypersurfaces with multiple fat points

Starting with the ground-breaking work of Cook II, Harbourne, Migliore and Nagel, there has been a lot of interest in unexpected hypersurfaces. In the last couple of months a considerable number of new examples and new phenomena has been observed and reported on. All examples studied so far had just one fat point. In this note we introduce a new series of examples, which establishes for the first time the existence of unexpected hypersurfaces with multiple fat points. The key underlying idea is to study Fermat-type configurations of points in projective spaces.

Weights of the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$ over a finite field

Grassmann codes are linear codes associated with the Grassmann variety $G(\ell,m)$ of $\ell$-dimensional subspaces of an $m$ dimensional vector space $\mathbb{F}_{q}^{m}.$ They were studied by Nogin for general $q.$ These codes are conveniently described using the correspondence between non-degenerate $[n,k]_{q}$ linear codes on one hand and non-degenerate $[n,k]$ projective systems on the other hand. A non-degenerate $[n,k]$ projective system is simply a collection of $n$ points in projective space $\mathbb{P}^{k-1}$ satisfying the condition that no hyperplane of $\mathbb{P}^{k-1}$ contains all the $n$ points under consideration. In this paper we will determine the weight of linear codes $C(3,8)$ associated with Grassmann varieties $G(3,8)$ over an arbitrary finite field $\mathbb{F}_{q}$. We use a formula for the weight of a codeword of $C(3,8)$, in terms of the cardinalities certain varieties associated with alternating trilinear forms on $\mathbb{F}_{q}^{8}.$ For $m=6$ and $7,$ the weight spectrum of $C(3,m)$ associated with $G(3,m),$ have been fully determined by Kaipa K.V, Pillai H.K and Nogin Y. A classification of trivectors depends essentially on the dimension $n$ of the base space. For $n\leq 8 $ there exist only finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\mathbb{\bar{F}}$ to $\mathbb{F}.$ This program is partially determined by Noui L. and Midoune N. and the classification of trilinear alternating forms on a vector space of dimension $8$ over a finite field $\mathbb{F}_{q}$ of characteristic other than $2$ and $3$ was solved by Noui L. and Midoune N. We describe the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$, where char $\mathbb{F}_{q}\neq 3.$ This fact we use to determine the weight of the $\mathbb{F}_{q}$-forms.

On a conjecture of Demailly and new bounds on Waldschmidt constants in [formula omitted

In the present note we prove a conjecture of Demailly for finite sets of sufficiently many very general points in projective spaces. This gives a lower bound on Waldschmidt constants of such sets. Waldschmidt constants are asymptotic invariants of subschemes receiving recently considerable attention [2], [1], [6], [11], [14].

The Aluffi algebra of the Jacobian of points in projective space: Torsion-freeness

The algebra in the title has been introduced by P. Aluffi. Let J⊂I be ideals in the commutative ring R. The (embedded) Aluffi algebra of I on R/J is an intermediate graded algebra between the symmetric algebra and Rees Algebra of the ideal I/J over R/J. A pair of ideals has been dubbed an Aluffi torsion-free pair if the surjective map of the Aluffi algebra of I/J onto the Rees algebra of I/J is injective. In this paper we focus on the situation where J is the ideal of points in general linear position in projective space and I is its Jacobian ideal.

Almost maximal growth of the Hilbert function

Let A=S/J be a standard artinian graded algebra over the polynomial ring S. A theorem of Macaulay dictates the possible growth of the Hilbert function of A from any degree to the next, and if this growth is the maximal possible then strong consequences have been given by Gotzmann. It can be phrased in terms of the base locus of the linear system defined by the relevant component(s) of J. If J is the artinian reduction of the ideal of a finite set of points in projective space then this maximal growth for A was shown by Bigatti, Geramita and the second author to imply strong geometric consequences for the points. We now suppose that the growth of the Hilbert function is one less than maximal. This again has (not as) strong consequences for the base locus defined by the relevant component. And when J is the artinian reduction of the ideal of a finite set of points in projective space, we prove that almost maximal growth again forces geometric consequences.

Counting points of given height that generate a quadratic extension of a function field

Let K be a finite algebraic extension of the field of rational functions in one indeterminate over a finite field and let [Formula: see text] denote an algebraic closure of K. We count points in projective space [Formula: see text] with given height and generating a quadratic extension of K. If n &gt; 2, we derive an asymptotic estimate for the number of such points as the height tends to infinity. Such estimates are analogous to previous results of Schmidt where the field K is replaced by the field of rational numbers ℚ.

Dimensional Differences Between Faces of the Cones of Nonnegative Polynomials and Sums of Squares:

We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases where there exist nonnegative polynomials that are not sums of squares. The gaps occur generically, they are not the product of selecting special faces of the cones. For ternary forms and quaternary quartics, we completely characterize when these differences are observed. Moreover, we provide an explicit description for these differences in the two smallest cases, in which the cone of nonnegative polynomials and the cone of sums of squares are different. Our results follow from more general results concerning the relationship between the second ordinary power and the second symbolic power of the vanishing ideal of points in projective space.

On the regularity of configurations of${\bf F}_q$-rational points in projective space

On the regularity of configurations of${\bf F}_q$-rational points in projective space