# Mark Huibregtse

## Professor of Mathematics

Class of 1964 Term Professor, 2009-2014

**Office: ** Harder Hall 216**Telephone: ** (518) 580-5289**Fax: ** (518) 580-5295 or (518) 580-5936**Email #1: **mhuibreg@skidmore.edu**Email #2: **huibregm@member.ams.org**Personal Web Page: **Mark Huibregtse

Professor Huibregtse teaches mathematics and computer science courses in the department. He received his Ph.D. in mathematics from the Massachusetts Institute of Technology and completed a Master’s-level program at the Institute for Retraining in Computer Science at Clarkson University. His research specialty is algebraic geometry, an area in which he has published a number of papers. Prof. Huibregtse has been active, along with other colleagues in the department, in trying to improve pedagogy through appropriate use of technology and alternative modes of learning and teaching. In 1996, Prof. Huibregtse completed five years of service as chair of the department. Outside Skidmore, Prof. Huibregtse is a spouse, parent, and member of the Methodist Church's chancel choir.

### Education

- Institute for Retraining in Computer Science (IFRICS), Clarkson University, Potsdam, NY, June 1985 - August 1986 (involved two intensive two-month summer sessions with an intervening yearlong project).
- Massachusetts Institute of Technology, Cambridge , MA , September 1971 - September 1976. Ph. D. in Mathematics. Thesis: The Albanese Mapping for the Hilbert Scheme of an Algebraic Surface, directed by Arthur P. Mattuck.
- Haverford College , Haverford , PA , September 1967 - May 1971. B. A., Magna cum Laude with High Honors in Mathematics, Phi Beta Kappa.

### Teaching Experience

- Skidmore College , Saratoga Springs , NY , September 1977 - present. Current title: Professor of Mathematics, and holder of the Class of 1964 Term Professorship (appointment from June 2009 – May 2014). Courses taught include: quantitative reasoning, pre-calculus, calculus, linear algebra, discrete mathematics, “bridge” to advanced mathematics, advanced calculus, real analysis, modern and differential geometry, algebraic curves, differential equations, number theory, BASIC programming, computer science I and II (using Pascal and C++), data structures, theory of computation, history of non-Euclidean geometry. Served as Department Chair from fall 1991 through spring 1996.
- University of Connecticut , Storrs , CT , September 1976 - May 1977. Visiting Assistant Professor of Mathematics. Taught calculus and linear algebra.
- Massachusetts Institute of Technology, Cambridge , MA . Served as a teaching assistant in calculus and elementary statistics while a graduate student (dates given above).

### Publications

*The Albanese Mapping for a Punctual Hilbert Scheme: I. Irreducibility of the Fibers*, Transactions of the American Mathematical Society 251 (1979), 267-285. MR 80g:14009.*The Albanese Mapping for a Punctual Hilbert Scheme: II. Symmetrized Differentials and Singularities*, Transactions of the American Mathematical Society 274 (1982), 109-140. MR 84h:14009.*Translation by Torsion Points and Rational Equivalence of 0-cycles on Abelian Varieties*, Rocky Mountain Journal of Mathematics 18 (1988), 147-155. MR 89j:14005.*Local Properties of Secant Varieties in Symmetric Products, Part I*, with Trygve Johnsen, Transactions of the American Mathematical Society 313 (1989), 187-204. MR 89i:14025.*Quotients of G-Stable Closed Subschemes, Cartesian Diagrams, and Closed Immersions*, Rocky Mountain Journal of Mathematics 25 (1995), issue #4, 1357-1384. MR 97a:14048.*Some Amplifications and a Correction of Iarrobino’s Construction of Local Parameters on Hilb^n_{P^2_k}*, Rocky Mountain Journal of Mathematics 29 (1999), issue #4, 1321-1361. MR 2001c:14008.*A Description of Certain Affine Open Subschemes that Form an Open Covering of Hilb^n_{A^2_k}*, Pacific Journal of Mathematics 204 (2002), issue #1, 97-143. MR 1 905 194M.*On Ellingsrud and Strømme’s cell decomposition of Hilb^{n}_{A^2_C}*, Inventiones mathematicae 160 (2005), 165-172. MR2129711.*An elementary construction of the multigraded Hilbert scheme of points*, Pacific Journal of Mathematics 223 (2006), no. 2, 269-315. MR2221028.*Some syzygies of the generators of the ideal of a border basis scheme*, Collectanea Mathematica 62 (2011), issue #3, pages 341-366. Initial publication is via the Web: http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s13348-010-0023-7 . Posted on the arXiv preprint server at arXiv:0908.0509. MR2825716.*Some elementary components of the Hilbert scheme of points*, Rocky Mountain Journal of Mathematics 47 (2017), issue #4, pages 1169-1225. DOI 10.1216/RMJ-2107-4-1169. Posted on the arXiv preprint server at arXiv:1407.1440v2 [math.AG].*Envelope curves and equidistant sets*, with Adam Winchell (Skidmore Class of 2016), Involve, a Journal of Mathematics 9 (2016), no. 5, 839-856. DOI 10.2140/involve.2016.9.839.

### Unpublished Manuscripts

*The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space*, written in summer 2004, revised spring 2005, and posted to the arXiv preprint server on 28 June 2005 (arXiv:math/0506575).

### Research Presentations

*Product Schemes with Group Action and Secant Sheaves on Quotients*, AMS Contributed Paper Session, Orono , ME , August 1991.*Quotients of G-Stable Subschemes: Questions of Embedding and Cartesian-ness*, a series of two talks given to the Capital Region Algebra and Number Theory Seminar, 29 October and 12 November 1991, Union College.*Quotients of G-Stable Subschemes, Cartesian Diagrams, and Embeddings*, AMS Session on Algebraic Geometry and Applied Group Theory, San Antonio, TX, January 1993.*Symmetrized Differentials on Hilbert Schemes of Points*, preliminary report, AMS Session on Algebraic Geometry, Cincinnati , OH , January 1994.*Symmetrized Differentials and Secant Sheaves on Hilbert Schemes of Points*, AMS Session on Algebraic Geometry, San Francisco, CA, January 1995.*An expanded version of the previous talk was presented to the CRANTS seminar at Union College*, March 1995.*The First Degeneracy Locus of the Map Sec(\Omega^1_X) à Sec(\Omega^1_{H^n_X}) for an Algebraic Surface X*, AMS Session on Commutative Algebra and Algebraic Geometry, Orlando, FL, January 1996.*On the Tangent Space at a k-point of Hilb^n_X for an Algebraic Surface X*, AMS Session on Algebraic Geometry, San Diego, CA, January 1997.*An Introduction to Hilb^n_{A^2_k} and to Some of its Local Charts*, Geometry, Algebra, and Topology (GAT) Seminar, Texas A&M University , 24 November 1997.*Some Amplifications and a Correction of Iarrobino’s Construction of Local Parameters on Hilb^n_{P^2_k}*, AMS Session on Algebraic Geometry and Fields, Baltimore , MD , January 1998.*A Description of Certain Affine Open Subschemes That Form an Open Covering of Hilb^n_{A^2_k}*, AMS Session on Algebraic Geometry, Joint AMS-MAA Winter Meetings at San Antonio, TX, 15 January 1999.*Open subschemes of Hilb^n_{A^2_k} that are isomorphic to A^{2n}_k*, AMS Session on Algebraic Geometry, Joint AMS-MAA Winter Meetings at Washington , DC , 19 January 2000.*An elementary construction of certain affine open subschemes of the Hilbert scheme of n points of the affine plane (Preliminary report)*, AMS Session on Algebraic Geometry and Linear Algebra, Joint AMS-MAA Winter Meetings at New Orleans, LA, 13 January 2001.*A description of certain affine open subschemes that form an open covering of the Hilbert scheme of n points of the affine plane*, a series of two talks presented to the CRANTS seminar meeting at Union College , 7 and 21 March 2001.*An elementary construction of an open affine covering of the Hilbert scheme of points of an affine space*, AMS Special Session on Hilbert Schemes, AMS Regional Meeting #979, Northeastern University , Boston , MA , 6 October 2002.*On Ellingsrud and Strømme's explicit description of a cell decomposition of the Hilbert scheme of points of the affine plane*, AMS Session on Algebraic Geometry and Algebra, Joint AMS-MAA Winter Meetings in Baltimore, MD, 17 January 2003.*The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space*, AMS Session on Algebra, Joint AMS-MAA Winter Meetings in Atlanta , GA , Wednesday, 5 January 2005.*An elementary construction of the Hilbert scheme of points of an affine space*, Union College Occasional Mathematics Conference, 3 December 2005.*Smooth monomial-ideal points of the Hilbert scheme of points of an affine space*, AMS Session on Algebraic Geometry, Joint AMS-MAA Winter Meetings in San Antonio , TX , 13 January 2006.*An elementary construction of the Hilbert scheme of points of an affine space*, presented to CRANTS seminar at Skidmore on 6 December 2006.*Some syzygies of the generators of the ideal of a border basis scheme*, AMS Session on Algebraic Geometry and K-Theory, Joint AMS-MAA Winter Meetings in Washington , DC , 8 January 2009.*Some Elementary Components of the Hilbert Scheme of Points*, AMS Session on Algebraic Varieties, Moduli Spaces, and Singularities, AMS-MAA Joint Mathematics Meetings in San Diego, CA, 12 January 2013.*A generalization of Emsalem and Iarrobino's construction of elementary components of the Hilbert scheme of points*, AMS Special Session on Commuting Nilpotent Matrices and the Hilbert Scheme, AMS Regional Meeting # 1088 at Chestnut Hill, MA, Sunday, 7 April 2013, 10:30-10:50 a.m., talk # 1088-14-89.

### Other Presentations

*Choosing the k-th Largest Element: a k-th Study in Algorithm Design and Analysis*, Skidmore Mathematics and Computer Science Colloquium, 17 November 1987.*ISETL and Mathematical Programming: Why Would I Settle For Less?*, Skidmore Mathematics and Computer Science Colloquium, 16 March 1989.*The Rise and Fall of Euclidean Geometry*, MAA Special Session on Using History in the Teaching of Mathematics, San Francisco, CA, January 1991.*Served as panelist for a discussion: What is the Mathematical Research Experience?*, Regional Geometry Institute, Amherst College , July 1992.*An Explicit Computation of an Invariant Subring of a Polynomial Ring*, Hudson River Undergraduate Mathematics Conference, Siena College , April 1994.*Demonstrating Space-Filling Curves with Derive: A Cautionary Tale?*, Hudson River Undergraduate Mathematics Conference, Siena College , April 1995.*Mathematica Programming*, presented as part of a departmental workshop on Mathematica, 31 May 2001.*The Banach-Tarski Paradox I: Preliminaries*, Hudson River Undergraduate Mathematics Conference, Union College , April 2009. (This was an introduction to the talk by my thesis student Rozalina Zlatareva, titled The Banach-Tarski Paradox II: Outline of Proof.)*The Merkle-Hellman Cryptosystem*, Hudson River Undergraduate Mathematics Conference, Keene State University, April 2010.- Equidistant sets of curves, Hudson River Undergraduate Mathematics Conference, St. Michael’s College, Colchester, VT, April 2016.