Publications (56)
Subdivision and spline spaces
Hal Schenck, Tatyana Sorokina
A standard construction in approximation theory is mesh refinement. For a simplicial or polyhedral mesh D in R^k, we study the subdivision D' obtained by subdividing a maximal cell…
Splines on the Alfeld split of a simplex and type A root systems
Hal Schenck
Alfeld introduced a subdivision AS(n) of an n-simplex, generalizing the Clough-Tocher split of a triangle. A formula for the dimension of the spline space C^r_k(AS(n)) was conjectu…
The method of shifted partial derivatives cannot separate the permanent from the determinant
Klim Efremenko, J. M. Landsberg, Hal Schenck +1
The method of shifted partial derivatives was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this met…
A case study in bigraded commutative algebra
David Cox, Alicia Dickenstein, Hal Schenck
We study the commutative algebra of three bihomogeneous polynomials p_0,p_1,p_2 of degree (2,1) in variables x,y;z,w, assuming that they never vanish simultaneously on P^1 x P^1. U…
Resonance varieties via blowups of P^2 and scrolls
Hal Schenck
Conjectures of Suciu relate the fundamental group of the complement M = C^n\A of a hyperplane arrangement A to the first resonance variety of H^*(M,Z). We describe a connection bet…
On minimal free resolutions of sub-permanents and other ideals arising in complexity theory
Klim Efremenko, J. M. Landsberg, Hal Schenck +1
We compute the linear strand of the minimal free resolution of the ideal generated by k x k sub-permanents of an n x n generic matrix and of the ideal generated by square-free mono…
The Weak Lefschetz Property and powers of linear forms in K[x,y,z]
Hal Schenck, Alexandra Seceleanu
We show that an Artinian quotient of K[x, y, z] by an ideal I generated by powers of linear forms has the Weak Lefschetz property. If the syzygy bundle of I is semistable this foll…
Algebraic properties of Hermitian sums of squares, II
Jennifer Brooks, Dusty Grundmeier, Hal Schenck
We study real bihomogeneous polynomials in complex variables for which is the squared norm of a holomorphic polynomial mapping. Such polyn…
Polynomial interpolation in higher dimension: from simplicial complexes to GC sets
Nathan Fieldsteel, Hal Schenck
Geometrically characterized (GC) sets were introduced by Chung-Yao in their work on polynomial interpolation in R^d. Conjectures on the structure of GC sets have been proposed by G…
Geometry of Wachspress surfaces
Corey Irving, Hal Schenck
Let P_d be a convex polygon with d vertices. The associated Wachspress surface W_d is a fundamental object in approximation theory, defined as the image of the rational map w_d fro…
The Vacuum Moduli Space of the Minimal Supersymmetric Standard Model
Yang-Hui He, Vishnu Jejjala, Brent D. Nelson +2
A starting point in the study of the minimal supersymmetric Standard Model (MSSM) is the vacuum moduli space, which is a highly complicated algebraic variety: it is the image of an…
Quaternary quartic forms and Gorenstein rings
Gregorz Kapustka, MichaÅ Kapustka, Kristian Ranestad +3
A quaternary quartic form, a quartic form in four variables, is the dual socle generator of an Artinian Gorenstein ring of codimension and regularity 4. We present a classification…
The Likelihood Correspondence
Thomas Kahle, Hal Schenck, Bernd Sturmfels +1
An arrangement of hypersurfaces in projective space is strict normal crossing (SNC) if and only if its Euler discriminant is nonzero. We study the critical loci of arbitrary Lauren…
The algebra of splines: duality, group actions and homology
Martina Lanini, Hal Schenck, Julianna Tymoczko
This survey gives an overview of three central algebraic themes related to the study of splines: duality, group actions, and homology. Splines are piecewise polynomial functions of…
Derivation modules of orthogonal duals of hyperplane arrangements
Joseph P. S. Kung, Hal Schenck
Let A be an n by d matrix having full rank n. An orthogonal dual A^{\perp} of A is a (d-n) by d matrix of rank (d-n) such that every row of A^{\perp} is orthogonal (under the usual…
The simplest minimal free resolutions in
Nicolás Botbol, Alicia Dickenstein, Hal Schenck
We study the minimal bigraded free resolution of an ideal with three generators of the same bidegree, contained in the bihomogeneous maximal ideal $ \langle s,t\rangle \cap \langle…
Tensor product surfaces and linear syzygies
Eliana Duarte, Hal Schenck
Let U be a basepoint free four-dimensional subpace of the space of sections of bidegree (a,b) on X = P^1 x P^1, with a and b at least 2. The sections corresponding to U determine a…
Toric Hirzebruch-Riemann-Roch via Ishida's theorem on the Todd genus
Hal Schenck
We give a simple proof of the Hirzebruch-Riemann-Roch theorem for smooth complete toric varieties, based on Ishida's result that the Todd genus of a smooth complete toric variety i…
Holonomy Lie algebras and the LCS formula for subarrangements of A_n
Paulo Lima-Filho, Hal Schenck
If X is the complement of a hypersurface in P^n, then Kohno showed that the nilpotent completion of the fundamental group is isomorphic to the nilpotent completion of the holonomy…
Equivariant Chow cohomology of nonsimplicial toric varieties
Hal Schenck
For a toric variety X_P determined by a rational polyhedral fan P in a lattice N, Payne shows that the equivariant Chow cohomology of X_P is the Sym(N)--algebra C^0(P) of integral…
Calabi-Yau threefolds in and Gorenstein rings
Hal Schenck, Mike Stillman, Beihui Yuan
A projectively normal Calabi-Yau threefold has an ideal which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals hav…
The Orlik-Terao algebra and 2-formality
Hal Schenck, Stefan Tohaneanu
The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement A in C^n; it is the quotient of an exterior algebra E(V) on |A| generators. Orlik and…
Toric surface codes and Minkowski sums
John Little, Hal Schenck
Toric codes are evaluation codes obtained from an integral convex polytope and finite field $\F_q$. They are, in a sense, a natural extension of Reed-Solomon codes…
Recent developments and open problems in linear series
Thomas Bauer, Cristiano Bocci, Susan Cooper +11
In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of i…
Algebraic methods in approximation theory
Hal Schenck
This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modelin…
Subarrangements of type A: the weak Lefschetz property of the Artinian Orlik-Terao algebra
Nicholas Gaubatz, Hal Schenck
In 1994, Orlik and Terao introduced a commutative Artinian analog S/I(A) of the Orlik-Solomon algebra of a hyperplane arrangement A to answer a question of Aomoto. A central topic…
Algebraic aspects of homogeneous Kuramoto oscillators
Heather Harrington, Hal Schenck, Mike Stillman
We investigate algebraic and topological signatures of networks of coupled oscillators. Translating dynamics into a system of algebraic equations enables us to identify classes of…
Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring
Hal Schenck
We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and monomial ideals to obtain a f…
Codes from surfaces with small Picard number
John Little, Hal Schenck
Extending work of M. Zarzar, we evaluate the potential of Goppa-type evaluation codes constructed from linear systems on projective algebraic surfaces with small Picard number. Put…
Free curves, Eigenschemes and Pencils of curves
Roberta Di Gennaro, Giovanna Ilardi, Rosa Maria Mirò-Roig +2
Let . A reduced plane curve is if its associated module of tangent derivations is a free -module, or equivale…
Quadratic Gorenstein rings and the Koszul property I
Matthew Mastroeni, Hal Schenck, Mike Stillman
Let be a standard graded Gorenstein algebra over a field presented by quadrics. Conca, Rossi, and Valla have shown that such a ring is Koszul if or if…
The Weak Lefschetz property for quotients by Quadratic Monomials
Juan Migliore, Uwe Nagel, Hal Schenck
In [MMR], MichaÅek--Miró-Roig give a beautiful geometric characterization of Artinian quotients by ideals generated by quadratic or cubic monomials, such that the multiplication…
Few smooth d-polytopes with n lattice points
Tristram Bogart, Christian Haase, Milena Hering +6
We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a…
Logarithmic vector fields for quasihomogeneous curve configurations in P^2
Hal Schenck, Hiroaki Terao, Masahiko Yoshinaga
Let A be a union of smooth plane curves C_i, such that each singular point of A is quasihomogeneous. We prove that if C is a smooth curve such that each singular point of A U C is…
Stratifying multiparameter persistent homology
Heather A. Harrington, Nina Otter, Hal Schenck +1
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a…
Betti Numbers and Degree Bounds for Some Linked Zero-Schemes
Leah Gold, Hal Schenck, Hema Srinivasan
In their paper on multiplicity bounds (1998), Herzog and Srinivasan study the relationship between the graded Betti numbers of a homogeneous ideal I in a polynomial ring R and the…
Local cohomology of logarithmic forms
Graham Denham, Hal Schenck, Mathias Schulze +2
Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian scheme of Y, homological invariants derived from logarithmic differential forms alon…
Hyperplane Arrangements: Computations and Conjectures
Hal Schenck
This paper provides an overview of selected results and open problems in the theory of hyperplane arrangements, with an emphasis on computations and examples. We give an introducti…
Quadratic Gorenstein rings and the Koszul property II
Matthew Mastroeni, Hal Schenck, Mike Stillman
A question of Conca, Rossi, and Valla asks whether every quadratic Gorenstein ring of regularity three is Koszul. In a previous paper, we use idealization to answer their quest…
Inverse systems, Gelfand-Tsetlin patterns and the weak Lefschetz property
Brian Harbourne, Hal Schenck, Alexandra Seceleanu
Migliore-Miró-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,x_4] generated by powers of li…
Nets in and Alexander Duality
Nancy Abdallah, Hal Schenck
A net in is a configuration of lines and points satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigro…
Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four
Nancy Abdallah, Hal Schenck
In 1978, Stanley constructed an example of an Artinian Gorenstein (AG) ring with non-unimodal -vector . Migliore-Zanello later showed that for regularity $r=…
The Module of Logarithmic p-forms of a Locally Free Arrangement
Mircea Mustata, Hal Schenck
For an essential, central hyperplane arrangement A in V=k^{n+1}, we show that Ω^1(A) (the module of logarithmic one forms with poles along A) gives rise to a locally free sheaf on…
Local Complete Intersections in P^2 and Koszul Syzygies
David Cox, Hal Schenck
We study the syzygies of a codimension two ideal I = <f_1,f_2,f_3> in k[x,y,z]. Our main result is that the module of syzygies vanishing (scheme-theoretically) at the zero locus Z…
Bernstein-Gelfand-Gelfand meets geometric complexity theory: resolving the 2 x 2 permanents of a 2 x n matrix
Fulvio Gesmundo, Hang, Huang +2
We describe the minimal free resolution of the ideal of subpermanents of a generic matrix . In contrast to the case of determinants, the $…
The Hessian polynomial and the Jacobian ideal of a reduced hypersurface in
Laurent Busé, Alexandru Dimca, Hal Schenck +1
For a reduced hypersurface of degree , the Castelnuovo-Mumford regularity of the Milnor algebra is well understood when is smooth, as…
Cayley-Bacharach and evaluation codes on complete intersections
Leah Gold, John Little, Hal Schenck
In recent work, J. Hansen uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in the proje…
Syzygies of projective toric varieties
Hal Schenck, Gregory G. Smith
This paper has been subsumed by math.AG/0502240
Linear systems on a special rational surface
Hal Schenck
We study the Hilbert series of a family of ideals J_Ïgenerated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question…
Vacuum Geometry of the Standard Model
Yang-Hui He, Vishnu Jejjala, Brent D. Nelson +2
Vacuum structure of a quantum field theory is a crucial property. In theories with extended symmetries, such as supersymmetric gauge theories, the vacuum is typically a continuous…
Betti tables forcing failure of the Weak Lefschetz Property
Sean Grate, Hal Schenck
We study the Artinian reduction of a configuration of points , and the relation of the geometry of to Lefschetz properties of . Migliore initia…
High rank linear syzygies on low rank quadrics
Hal Schenck, Mike Stillman
We study the linear syzygies of a homogeneous ideal I in a polynomial ring S, focusing on the graded betti numbers b_(i,i+1) = dim_k Tor_i(S/I, k)_(i+1). For a variety X and diviso…
A new bound for smooth spline spaces
Hal Schenck, Mike Stillman, Beihui Yuan
For a planar simplicial complex Delta contained in R^2, Schumaker proved that a lower bound on the dimension of the space C^r_k(Delta) of planar splines of smoothness r and polynom…
Syzygies, multigraded regularity and toric varieties
Milena Hering, Hal Schenck, Gregory G. Smith
Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B_1, ..., B_k on X an…
Syzygies and singularities of tensor product surfaces of bidegree (2,1)
Hal Schenck, Alexandra Seceleanu, Javid Validashti
Let U be a basepoint free four-dimensional subspace of the space of sections of O(2,1) on P^1 x P^1. The sections corresponding to U determine a regular map p_U: P^1 x P^1 --> P^3.…
Freeness of Conic-Line Arrangements in
Hal Schenck, Stefan O. Tohaneanu
Let be a collection of smooth rational plane curves. We prove that the addition-deletion operation used in the study of h…