Juan C. Migliore

Associate Chair

Professor

B.A., Haverford College, 1978
Ph.D, Brown University, 1983

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, 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 recent monograph listed second below.

A common theme in my research has been the use of deficiency modules as a tool to provide geometric and algebraic information about subschemes of projective space. The deficiency modules of a scheme arise as the cohomology of the ideal sheaf of the scheme. Most basically, they measure the failure of the scheme to be arithmetically Cohen-Macaulay, but they also give more subtle information. The deficiency modules also give important information about the liaison class of the scheme. I completed a book about liaison theory and deficiency modules, listed third below.

While I am still quite interested in liaison theory, most recently my research has shifted. I have been studying ideals generated by a general collection of homogeneous polynomials of given degrees, and in particular I have studied the Hilbert function and minimal free resolution of such an ideal. This is closely related to the so-called Strong Lefschetz property for graded Artinian ideals. Two of my most recent papers on these areas aare listed fourth and fifth below.

Selected Publications

• E. Ballico, G. Bolondi and Juan Migliore, The Lazarsfeld-Rao problem for liaison classes of two-codimensional subschemes of Pⁿ. 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 R. Miró-Roig, On the Minimal Free Resolution of n+1 Generic Forms, Trans. Amer. Math. Soc. 355 (2003), 1--36.
• T. Harima , J. Migliore, U. Nagel and J. Watanabe, The Weak and Strong Lefschetz Properties for Artinian K-Algebras, to appear in J. Algebra.

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