___________________________________________________________________ Hal Schenck

Hal Schenck

Mathematics Department, 1409 W. Green St
University of Illinois
Urbana, IL 61801
tel: 217-333-2229
fax: 217-333-9576

schenck@math.uiuc.edu

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Who am I?

I am an Associate Professor in the Mathematics Department at University of Illinois, and also at Texas A&M University (on leave). My research interests are in the areas of:

ocommutative algebra
oalgebraic geometry
odiscrete geometry and combinatorics
I'm especially interested in problems which can be studied from a computational standpoint, and in interactions of commutative algebra with problems in geometry and combinatorics, particularly applied problems. Among my current interests in applied algebraic geometry are surface modelling and coding theory. I also enjoy working on problems at the interface of discrete geometry and algebra; for example hyperplane arrangements (for an overview, you can watch my lecture at the MSRI introductory workshop on arrangements) and toric varieties. I've also worked on questions involving free resolutions, fatpoints, rational surfaces, Green's conjecture, stability and jump loci of vector bundles, and splines. Here's my resume.
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Information for students

In Fall 2008 I'm teaching Math 510, Algebraic Curves and Riemann Surfaces For graduate students interested in commutative algebra and algebraic geometry, here are a few thoughts on what to read and how to choose an advisor. For a turbo introduction to algebraic geometry, you can check out notes from a course in Computational Algebraic Geometry that I taught at Harvard in fall 2000. For more advanced topics, check out Joe Harris's Second course in Algebraic Geometry and David Eisenbud's notes on The Geometry of Syzygies. Below are webpages of former students and postdocs:
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Book projects

I'm currently at work on two book projects. David Cox, John Little and I are at work on a book on Toric Varieties, and with Dan Cohen, Graham Denham, Mike Falk, Alex Suciu, Hiro Terao and Sergey Yuzvinsky, I'm writing a book on Complex Arrangements:Algebra, Geometry, Topology.
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Upcoming/Recent events

Some events of interest:
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Publications

o Betti numbers and degree bounds for some linked zero schemes, (joint with Leah Gold and Hema Srinivasan), Journal of Pure and Applied Algebra, 210, (2007), 481-491.
o A case study in bigraded commutative algebra, (joint with David Cox and Alicia Dickenstein), in "Syzygies and Hilbert Functions", edited by Irena Peeva, Lecture notes in Pure and Applied Mathematics 254, (2007), 67--112.
o Syzygies, multigraded regularity and toric varieties, (joint with Milena Hering and Gregory G. Smith), Compositio Mathematica, 142, (2006), 1499-1506.
o Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence, (joint with Alex Suciu), in Transactions of the A.M.S., 358, (2006), 2269-2289.
o Toric surface codes and Minkowski sums, (joint with John Little), SIAM Journal on Discrete Mathematics, 20, (2006), 999-1014.
o Derivation modules of orthogonal duals of hyperplane arrangements, (joint with Joseph P.S. Kung), in Journal of Algebraic Combinatorics, 24, (2006), 253-262.
o Cayley-Bacharach and evaluation codes on complete intersections, (joint with Leah Gold and John Little), in Journal of Pure and Applied Algebra, 196, (2005) p. 91-99.
o Linear systems on a special rational surface, in Mathematical Research Letters, 11, (2004) p. 697-714.
o Lattice polygons and Green's theorem, in Proceedings of the A.M.S., 132, (2004) p. 3509-3512.
o Elementary modifications and line configurations in P^2, in Commentarii Mathematici Helvetici, 78, (2003) p. 447-462.
o Local complete intersections in P^2 and Koszul syzygies, (joint with David Cox), in Proceedings of the A.M.S., 131, (2003) p. 2007-2014.
o Lower central series and free resolutions of hyperplane arrangements, (joint with Alex Suciu), in Transactions of the A.M.S., 354, (2002) p. 3409-3433.
o Cohomology vanishing and a problem in approximation theory, (joint with Peter Stiller), in Manuscripta Mathematica, 107, (2002) p. 43-58.
o The module of logarithmic p-forms of a locally free arrangement, (joint with Mircea Mustata), in Journal of Algebra, 241, (2001) p. 699-719.
o On a conjecture of Rose, (joint with John Dalbec), in Journal of Pure and Applied Algebra, 165, (2001) p. 151-154.
o A rank two vector bundle associated to a three arrangement, and its Chern polynomial, in Advances in Mathematics, 149, (2000) p. 214-229.
o Subalgebras of the Stanley-Reisner ring, in Discrete and Computational Geometry, 21, (1999) p. 551-556.
o Fat points, inverse systems, and piecewise polynomial functions, (joint with Anthony Geramita), in Journal of Algebra, 204, (1998) p. 116-128.
o A spectral sequence for splines, in Advances in Applied Mathematics, 19, (1997) p. 183-199.
o Local cohomology of bivariate splines, (joint with Mike Stillman), in Journal of Pure and Applied Algebra, 117-118, (1997) p. 535-548.
o A family of ideals of minimal regularity and the Hilbert series of C^r(\Delta), (joint with Mike Stillman), in Advances in Applied Mathematics, 19, (1997) p. 169-182.
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Preprints

o Freeness of Conic-Line arrangements in P^2, (joint with Stefan Tohaneanu), to appear in Commentarii Mathematici Helvetici.
o Efficient computation of resonance varieties via Grassmannians, (joint with Paulo Lima-Filho), to appear in Journal of Pure and Applied Algebra.
o Piecewise polynomials on polyhedral complexes, (joint with Terry Mcdonald), to appear in Advances in Applied Mathematics.
o Holonomy Lie algebras and the LCS formula for subarrangements of A_n, (joint with Paulo Lima-Filho).
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Computational mathematics

oMacaulay2: A program for computing in commutative algebra and algebraic geometry.
oCoCoA: Another option for computing in commutative algebra and algebraic geometry.
oPorta (and others): Programs for polyhedral geometry.
oSingular: A program for computations in local rings and singularities.
oSchubert: A program for enumerative geometry.
oSplinetool: Macaulay2 script which builds the chain complexes and allows computation of various homology modules associated to splines.
oArrangetool: Macaulay2 script for computations involving three arrangements.
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Sites I visit frequently

o ams web page
o siam web page
o New York times web page
o Algebraic Geometry preprints
oMSRI (Berkeley).
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