# Juan Migliore

*Professor*

## Education

B.A., Haverford College, 1978

Ph.D, Brown University, 1983

## Research Group

Algebra and Algebraic Geometry

## Research Area

Algebraic Geometry/Commutative Algebra

## Bio

**Research Interests**

In the first part of my career, my research has focused on liaison theory of subschemes of projective space, a specialty within the broader area of algebraic geometry and commutative algebra. I began by studying the case of codimension two subschemes, and the high point of that early part of my work is the paper listed first below. Later my research turned to questions in higher codimension, and in the more elusive case of Gorenstein liaison. The high point of this part of my work is the monograph listed second below. I also completed a book about liaison theory and so-called deficiency modules, listed third below. The fourth and fifth papers listed below proved important special cases of a still-open question in liaison theory.

While I am still quite interested in liaison theory, over the last couple of decades my research has shifted. An important theme has been the study of Hilbert functions of various kinds of algebras, as well as their graded Betti numbers. In particular, I have studied the cases of level (possibly monomial) algebras, algebras defined by ideals of generic forms, and Gorenstein algebras. The papers listed sixth through tenth below are representative of this kind of study. A related topic which has occupied a fair amount of my time in the last decade or so is the so-called Weak Lefschetz Property for artinian algebras. Two of my papers in this area are listed eleventh and twelfth below. The last paper listed below is opening the way for a new area of study that I’m currently very interested in pursuing.

My personal page contains a link to my complete list of publications.

**Selected Publications**

- E. Ballico, G. Bolondi and Juan Migliore. The Lazarsfeld-Rao problem for liaison classes of two-codimensional subschemes of P^n. Amer. J. of Math., 113:117-128, 1991.
- J. Kleppe, J. Migliore, R.M. Miró-Roig, U. Nagel and C. Peterson. Gorenstein Liaison, Complete Intersection Liaison Invariants and Unobstructedness. Memoirs of the Amer. Math. Soc., 154, 2001.
- J. Migliore. “Introduction to Liaison Theory and Deficiency Modules. Birkhauser, Progress in Mathematics 165, 1998; 224 pp. Hardcover, ISBN 0-8176-4027-4.
- J. Migliore and U. Nagel. Monomial ideals and the Gorenstein liaison class of a complete intersection. Compositio Math., 133:25-36, 2002.
- J. Migliore and U. Nagel. Glicci ideals. Compositio Mathematica 149:1583-1591, 2013.
- A. Bigatti, A.V. Geramita, and J. Migliore. Geometric consequences of extremal behavior in a theorem of Macaulay. Trans. Amer. Math. Soc. 346:203-235, 1994.
- J. Migliore and U. Nagel. Reduced arithmetically Gorenstein schemes and Simplicial Polytopes with maximal Betti numbers. Adv. Math 180:1-63, 2003.
- A.V. Geramita, T. Harima, J. Migliore and Y. Shin. The Hilbert function of a level algebra. Memoirs of the Amer. Math. Soc., 186, 2007.
- J. Migliore, U. Nagel and F. Zanello. A characterization of Gorenstein Hilbert functions in codimension four with small initial degree. Math. Res. Lett. 15:331-349, 2008.
- M. Boij, J. Migliore, R. Miró-Roig, U. Nagel and F. Zanello. The shape of a pure O-sequence. Memoirs Amer. Math. Soc. vol. 218, No. 2024, July, 2012.
- T. Harima , J. Migliore, U. Nagel and J. Watanabe. The Weak and Strong Lefschetz Properties for Artinian K-Algebras. J. Algebra, 262:99-126, 2003.
- J. Migliore, R. Miró-Roig and U. Nagel. Monomial ideals, almost complete intersections and the Weak Lefschetz Property. Trans. Amer. Math. Soc., 363:229-257, 2011.
- D. Cook II, B. Harbourne, J. Migliore and U. Nagel. Line arrangements and configurations of points with an unexpected geometric property. Accepted for publication in Compositio Math.

**Email:** jmiglior@nd.edu

**Phone:** 574-631-7345

**Fax:** 574-631-6579

**Office:** 236 Hayes-Healy Bldg