Research

Abstract: The purpose of this work is to pursue classification of geproci sets. Specifically we classify [m,n]-geproci sets which consist of m=4 points on each of n skew lines, assuming the skew lines have two transversals in common. We show that in this case n≤6. Moreover we show that all geproci sets of this type are contained in the standard construction for m=4 introduced in arXiv:2209.04820. Finally, we propose a conjectural representation for all geproci sets of this type, irrespective of the number m of points on each skew line.

Abstract: Geproci sets of points in ℙ3 are sets whose general projections to ℙ2 are complete intersections. The first nontrivial geproci sets came from representation theory, as projectivizations of the root systems D4 and F4. In most currently known cases geproci sets lie on very special unions of skew lines and are known as half grids. For this important class of geproci sets we establish fundamental connections with combinatorics, which we study using methods of algebraic geometry and commutative algebra. As a motivation for studying them, we first prove Theorem A: for a nondegenerate (a,b)-geproci set Z with d being the least degree of a space curve C containing Z, that if d≤b, then C is a union of skew lines and Z is either a grid or a half grid. We next formulate a combinatorial version of the geproci property for half grids and prove Theorem B: combinatorial half grids are geproci in the case of sets of a points on each of b skew lines when a≥b−1≥3. We then introduce a notion of combinatorics for skew lines and apply it to the classification of single orbit combinatorial half grids of m points on each of 4 lines. We apply these results to prove Theorem C, showing, when n≫m, that half grids of m points on n lines with two transversals must be very special geometrically (if they even exist). Moreover, in the case of skew lines having two transversals, our results provide an algorithm for enumerating their projective equivalence classes. We conjecture there are (m2−1)/12 equivalence classes of combinatorial [m,4]-half grids in the two transversal case when m>2 is prime.

Abstract: In this short note we develop new methods toward the ultimate goal of classifying geproci sets in P3. We apply these method to show that among sets of 16 points distributed evenly on 4 skew lines, up to projective equivalence there are only two distinct geproci sets. We give different geometric distinctions between these sets. The methods we develop here can be applied in a more general set-up; this is the context of follow-up work in progress.

Abstract: This work was motivated by wanting to explore the existence of nondegenerate finite sets of points Z ⊂ P3 whose projection from a general point to a plane is a complete intersection.  

We call a set of points Z⊂ P3  an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees  a and b.  Examples which we call grids have been known since 2011.
Here, for any 4 ≤ a ≤  b, we construct nongrid nondegenerate (a,b)-geproci sets in a systematic way. We also show that the only such example with a=3 is a (3,4)-geproci set coming from the D_4 root system, and we describe the D_4 configuration in detail. 

We also consider the question of the equivalence (in various senses) of geproci sets, as well as which sets occur over the reals, and which cannot. We identify several additional examples of geproci sets with interesting properties. We also explore  the relation between unexpected cones and geproci sets and introduce the notion of d-Weddle schemes arising from special projections of finite sets of points. 

This work initiates the exploration of new perspectives on classical areas of geometry. We formulate and discuss a range of open problems in the final chapter.

Abstract: The contraction of one or more edges in a graph $G$ produces a new graph $G'$ whose total Betti numbers are, in general, not related to those of $G$. In this paper, we study contractions for which $G$ has either total Betti numbers greater or equal to those of  $G'$, i.e., $\beta_i(\K[G])\ge \beta_i(\K[G']), i\ge 0$, or the same total Betti numbers as $G'$, that is, $\beta_i(\K[G])= \beta_i(\K[G']), i\ge 0$.

Abstract: A k-configuration of type (d1,...,ds) is a set of points in P2 that has a number of algebraic and geometric properties. All the graded Betti numbers and the Hilbert function of a k-configuration in P2 is determined by the type (d1,...,ds). However, the type alone is not enough to determine the Waldschmidt constant. In this paper, we compute the Waldschmidt constant of a standard k-configuration in P2  of type (a,b,c) with a≥1 except the type (2,3,5). In particular, we prove that the Waldschmidt constant of a standard k-configuration in P2 of type (1,b,c) with c≥2b+ 2 does not depend on c. 

Abstract: Given r>n general hyperplanes in P^n a star configuration of points is the set of all the n-wise intersection of the hyperplanes..  We introduce  contact star configurations, which are star configurations where all the hyperplanes are osculating to the same rational normal curve. In this paper we find a relation between this construction and Hadamard products of linear varieties. Moreover, we study the union of contact star configurations on a same conic in P^2, we prove that the union of two contact star configurations has a special h-vector and, in some cases, this is a complete intersection.

A classical kind of problem in algebraic geometry is to consider vanishing conditions on a linear system L, and to ask if the dimension of the resulting linear system is what one would expect based on the dimension of L and the specific conditions imposed. 

Abstract: If X⊂Pn is a reduced subscheme, we say that X admits an unexpected hypersurface of degree t for multiplicity m if the imposition of having 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 understand. 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 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. 

Abstract:This paper examines the Arithmetically Cohen-Macaulay (ACM) property for certain codimension 2 varieties in P1×P2 called sets of lines in P1×P2 (not necessarily reduced). We discuss some obstacles to finding a general characterization. We then consider certain classes of such curves, and we address two questions. First, when are they themselves ACM? Second, in a non-ACM reduced configuration, is it possible to replace one component of a primary (prime) decomposition by a suitable power (i.e. to "fatten" one line) to make the resulting scheme ACM? Finally, for our classes of such curves, we characterize the locally Cohen-Macaulay property in combinatorial terms by introducing the definition of a fully v-connected configuration. We apply some of our results to give analogous ACM results for sets of lines in P3. 

Abstract: Let I⊆R=K[x1,…,xn] be a toric ideal, i.e., a binomial prime ideal. We investigate when the ideal I can be "split" into the sum of two smaller toric ideals. For a general toric ideal I, we give a sufficient condition for this splitting in terms of the integer matrix that defines I. When I=I_G is the toric ideal of a finite simple graph G, we give additional splittings of I_G related to subgraphs of G. When there exists a splitting I=I1+I2 of the toric ideal, we show that in some cases we can describe the (multi-)graded Betti numbers of I in terms of the (multi-)graded Betti numbers of I1 and I2. 

Abstract: 

A famous theorem, due to Macaulay  and pointed out by Stanley, characterizes the numerical functions that are Hilbert functions of a standard graded k-algebra. A generalization of Macaulay’s theorem to multi-graded rings is an open problem.  In this work we generalize the Macaulay’s Theorem to the first significant case of bigraded algebras. 

Abstract: We classify the Hilbert functions of bigraded algebras in k[x1, x2, y1, y2] by introducing a numerical function called a Ferrers function. 

Abstract: The aim of this paper is to make a connection between design theory and algebraic geometry/commutative algebra. In particular, given any Steiner System S(t, n, v) we associate two ideals, in a suitable polynomial ring, defining a Steiner configuration of points and its Complement. We focus on the latter, studying its homological invariants, such as Hilbert Function and Betti numbers. We also study symbolic and regular powers associated to the ideal defining a Complement of a Steiner configuration of points, finding its Waldschmidt constant, regularity, bounds on its resurgence and asymptotic resurgence. We also compute the parameters of linear codes associated to any Steiner configuration of points and its Complement.

  Let H=(V,E) be a hypergraph. We say that a hypergraph  H'=(V', E') is a shadow of H if  

1. V'⊆  V; 

2. |E|=|E'| (same cardinality) and  e ⋂ V' ∈  E'  for each e ∈  E. 

Abstract: The aim of this paper is to study the associated primes of powers of squarefree monomial ideals. Hypergraphs and squarefree monomial ideals are strongly connected. The cover ideal J(H) of a hypergraph H is the intersection of the primes corresponding to the edges of H. We define the shadow of H as a certain set of smaller hypergraphs related to H. We then describe how the shadows of H preserve information about the associated primes of the powers of J(H). Some implications to the persistence property are studied. 

Abstract: For all integers 4≤r≤d, we show that there exists a finite simple graph G=Gr,d with toric ideal IG⊂R such that R/IG has (Castelnuovo-Mumford) regularity r and h-polynomial of degree d. To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and furthermore, this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O'Keefe that compares the depth and dimension of toric ideals of graphs. 

Take a positive integer, e.g.  3, and construct the sequences

• 3

• 1,2,3

• 1,1,2,1,2,3

• 1,1,1,2,1,1,2,1,2,3,

• ...

where, in each step, a number "n" is replaced by "1,..., n".  Hilbert functions and graded Betti numbers of the lex segments ideals can be computed by suitable truncation of these sequences.

Abstract: We introduce the fractal expansions, sequences of integers associated to a number. We show that these sequences characterize the O-sequences and encode some information on lex segment ideals. Moreover, we introduce a numerical functions called fractal functions, and we show that fractal functions can be used to classify the Hilbert functions of bigraded algebras. 

Let X be a finite collection of points in a multiprojective  space. It is interesting to describe the homological invariants of the coordinate ring of X. In particular, it is a subject of research to understand when X is arithmetically Cohen-Macaulay (ACM), i.e. when the coordinate ring is a Cohen-Macaulay ring. Since it is no longer the case (as it is in projective space) that a finite set of points is automatically ACM, the determination of whether or not the ACM property holds for a finite set draws on a combination of geometric, combinatoric, algebraic and numerical considerations. The Cohen-Macaulay question here is closely related to the Cohen-Macaulay question for unions of linear varieties in projective space, but it is a more manageable version of the problem than the case of arbitrary unions since only certain such unions correspond to finite sets in multiprojective spaces.

Abstract: In this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1 × ℙ1 and, more recently, in (ℙ1)^r. In ℙ1 × ℙ1 the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm × ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1 × ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting. 

A motivating problem in algebraic geometry and commutative algebra concerns multiprojective spaces. Given a finite collection of points X  in a multiprojective space it is interesting to describe the homological invariants of the quotient ring of X. An important property is whether the collection is arithmetically Cohen-Macaulay (ACM) or not, i.e. whether the quotient ring is a Cohen-Macaulay ring. It is no longer the case (as it is in projective space) that a finite set of points is automatically ACM. It is of interest to understand which finite sets of points are ACM. 

Abstract: We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially (P1)^n. A combinatorial characterization, the (⋆)-property, is known in P1×P1. We propose a combinatorial property, (⋆n), that directly generalizes the (⋆)-property to (P1)^n for larger n. We show that X is ACM if and only if it satisfies the (⋆n)-property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space. 

Abstract: We study the Hilbert function and the graded Betti numbers for “generic” linear quotients of Artinian standard graded algebras, especially in the case of Weak Lefschetz algebras. Moreover, we investigate a particular property of Weak Lefschetz algebras, the Betti Weak Lefschetz Property, which makes possible to completely determine the graded Betti numbers of a generic linear quotient of such algebras.