# Juan C. Migliore

Professor

B.A., Haverford College, 1978

Ph.D, Brown University, 1983

Email: Juan.C.Migliore.1@nd.edu

Office:236 Hayes-Healy

Phone: (574) 631-7345

Fax: (574) 631-6579

For additional information see Juan Migliore’s Personal Page.

**Research Interests**

My main research to date has been on liaison theory, 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 probably the paper listed first below. More recently, I have been very much interested in 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 work listed below proved an important special case of a still-open question in liaison theory.

While I am still quite interested in liaison theory, over the last decade my research has shifted somewhat. An important theme has been the study of Hilbert functions of various 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. See for instance the fifth and sixth references below. A related topic is the so-called Weak Lefschetz Property for artinian algebras. Two of my papers on this topic are listed seventh and eighth below.

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.
- 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.
- 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.

Please direct questions and comments to: Juan.C.Migliore.1@nd.edu