Annotated List of Publications (and some Preprints)

Ph.D. thesis Families of ideals in a power series ring in two variables, (Arthur Mattuck, advisor), M.I.T., 1970.


Uses standard bases techniques to parametrize ideals in $k\{ x,y\}$ of given Hilbert function $T$, showing that the family $Z(T)$ (all) is fibred by affine spaces of constant dimension over $G(T)$ (graded), which is a projective, variety, irreducible, covered by a finite number of affine spaces, of known dimension. The unattained goal had been to show the curvilinear conjecture, that any punctual scheme concentrated at a point of $P^2$ was the specialization of a family of curvilinear schemes, those with local Hilbert function $(1,1,...,1)$: this had been conjectured by Nagata, and had been studied by Hartshorne also. At the same time, Brian\c{c}on was developing independently similar results using standard basis theory from Hironaka. Brian\c{c}on went further and proved the curvilinear conjecture in about 1972, published in 1977 as "Description de Hilb^n(C{x,y}".Rather later Granger using a method of Gaffney generalized the result to order u specializes to all of order u+1 (in two variables).


1. Punctual Hilbert Schemes, BAMS 78 (1972), 810-823 (MR 46 #7235).


Announces some main results from thesis. In particular I determine the rational Picard group of $Hilb^n(P^2)$, the punctual Hilbert scheme of projective space. There is a divisor coming from the multiple points of multiplicities (2,1,....,1); but the locus over points of higher multiplicity has higher codimension than one.


2. Reducibility of the family of 0-dimensional schemes on a variety, Inventiones Math. 15 (1972), 72-77 (MR 46 #170).


Gave the first known example -- surprising at the time -- of a zero-dimensional scheme on $P^r, r=3$ that could not be smoothed, using a dimension argument. Independently, Mumford developed a similar argument while showing that certain "pathological" families of curves could not be smoothed. The argument demonstrates the existence of "generic" singularities -- corresponding to irreducible components of $Hilb^n(P^r), r\ge 3$ other than the component parametrizing smooth schemes, but does not give a specific example of one: rather, a suitable deformation of a nonsmoothable scheme must be "generic"


3. Number of generic singularities, Rice University Studies 59, Part 1 (1972), 49-52 (MR 49 #10693).


Discusses the topography of the punctual Hilbert scheme: since elementary components (concentrated at a single point) of Hilb^{n_i}(P^r$ (parametrizing degree-$n_i$ schemes) can combine together to make irreducible components of $Hilb^n(P^r), n=\sum n_i$, it follows that once one has two elementary components of a single Hilb^{n_1}(P^r), one obtains a near-exponential increase with n in the number of components of Hilb^n(P^r).


4. An algebraic fibre bundle over $P^1$ that is not a vector bundle, Topology 12 (1973) (MR 49 #5009)


Considering Hilbert functions $T=(1,1,...,1)$ of length n, one finds that $Z_T$, although an affine bundle with a "zero" section over $G_T$, can not be linearized, for n\ge 4.


5. Memoir:
   Punctual Hilbert Schemes, 111 p., A.M.S. Memoir #188, A.M.S., Providence, 1977 (MR 58 #5667)


I included most results from thesis, developed a characteristic p analog of Brian\c{c}on's irreducibility result, and began a theory about the variety $Grass(T)$ parametrizing vector spaces of degree-$j$ forms generating an ideal of given Hilbert functiion $T$: I determined the closure of $Grass(T)$ in $Grass(d,R_j)$.
Huibregste much later found an error in claimed local parameters for points of the Hilbert scheme. Huibregste found good parameters in some rather special cases; Haiman recently found a good set of local parameters around monomial ideals, a result that presumably can be extended.


6. (with Jacques Emsalem) Some zero-dimensional generic singularities: finite algebras having small tangent spaces, Compositio Math. 36 (1978), 145-188 (MR 81c #14004).


We develop a different method, the "small tangent space" method, to demonstrate that a given zero-dimensional scheme is generic, and give the first examples. Basically one shows that the vector space dimension of the tangent space Hom(I,R/I) to the Hilbert scheme, is equal to the dimension of the family of schemes having a give type and local Hilbert function; so there can be no deformation outside of the family. One uses that there is a grading on the tangent space, and one shows that there are no negative degree tangents, other than those corresponding to the trivial deformation, of moving the point of support on P^r. The method requires a calculation of Hom(I,R/I) for one example in each expected case. The calculation in Section 3.2 for the Gorenstein T=(1,4,4,1) case (taking several weeks in the climbing practice area of Fontainbleau - now it takes less than a second on a computer) has an error, that was noticed using a different argument by Buchweitz: the first Gorenstein $T=(1,r,r,1)$ case where the calculation works is r=6. Later, Shafarevitch showed that for (non-Gorenstein) Artin algebras $A=R/I$ Hilbert function $T=(1,r+1,k,0)$ -- determined by a generic $k$-codimension vector space of quadratic forms $I_2$, that for suitable r and for k in a middle third of values, that $A$ is a generic singularity.


7. (with Alan Altman and Steven L. Kleiman) Irreducibility of the compactified Jacobian, pp. 1-12 in Real and Complex Singularities, Oslo 1976, P. Holm, editor, Sijthoff & Noordhoff, Alphen aan den Rign, Netherlands, 1978 (MR 58 #16650).


My contribution was to show that the good results in the planar case did not extend to the case of curves in P^3, due to the trick of embedding a "generic" punctual singularity at one point of the curve, possible in P^3 (but not in P^2).


8. (with Joel Briancon) Dimension of the punctual Hilbert scheme, Journal of Algebra 55 (1978), 536-544 (MR 80 g #14013).


We proved that there is an upper bound on the dimension, of Hilb^n(P^r), of the form $cn^{2-2/r}$; by the earlier dimension examples, there is a lower bound of similar form. We in fact bound the dimension of the tangent space Hom(I,R/I); a general Borel fixed point argument shows that maxima must be attained at a monomial ideal; we conjectured that the monomial ideal with maximal tangent space dimension could be assumed to have the first monomials of each degree in alphabetic order as cobasis, but we were not able to show this. As a result, our proof became technically more difficult. Very recently, Sturmfels found a couterexample to the conjecture!


9. Deformations of zero-dimensional complete intersections, after M. Granger, T. Gaffney, with appendix: Global Hilbert Scheme $Hilb^QP^r$ after Gotzmann, pp. 92-105 in Proceedings of the Week of Algebraic Geoemtry, Bucharest, 1980, L. Badescu and H. Kurke, editors, Teubnertexte #40, B.G. Teubner, Leipzig, 1981 (MR 84j:14012).


The first part is developed further in the next article; the second was an attempt to make more well known G. Gotzmann's striking proof of a "Persistence Theorem", that if the ideal $I=(V)$ generated by a degree-j vector space of forms $V\subset R$, a polynomial ring, grows minimally in the next degree -- if $dim_k(R_1\cdot V)$ attains the minimum value given by Macaulay, given $dim_k V$, then it continues to grow minimally: that is, $dim_k R_2\cdot V$ is the Macaulay minimum, given $dim_k R_1\cdot V$, etc. I was interested in this as a Ph.D. student, David Berman and I had discussed the question, conjectured it would be true, but were not able to show it even in three variables. See the Appendix to Springer LNM #1721, below for more.


10. Deforming complete intersection Artin algebras., pp. 593-608 in Singularities, PSPM #40, part 1, American Math Society, Providence, RI, 1983.


A simple dimension calculation shows that the (graded) complete intersections concentrated at a point, in r-variables, r \ge 4, although smoothable, quickly become non-alignable: have no deformation to a curvilinear ideal, as the colength = degree increases. The article goes on to discuss several conjectures involving deformation of complete intersections, involving very small components of the Hilbert scheme -- which should, conjecturally, consist of "peelable" algebras that can be deformed to a smooth scheme in steps of peeling of a single smooth point. An appendix deals with several different matters involving vector spaces of forms that seemed less well known: with the possible Hilbert function of the ideal generated by a vector space of forms in two variables, with the closure of these Hilbert function strata, and with Wronskians.


11. Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. AMS 285 (1984),337-378.


This paper introduces compressed algebras,those having maximum Hilbert function, given the socle type: they generalize Peter Schenzel's extremal algebras, having a maximum order of defining ideal, given the socle degree. I determined the compressed Hilbert function - in the Gorenstein case this was also shown at about the same time by C. Greene, namely $H=H(r,j): H_i=max(r_i,r_{j-i})$, the obvious candidate, by symmetry.
The reason for studying compressed algebras is first, their extremal properties, but also because - both in graded and in nongraded cases, they can be simply parametrized, using generators of Macaulay's inverse systems. Thus the study of graded Gorenstein algebra quotients $A=R/I$ having a given Hilbert function $H=H(A)$, is the same as that of the family of degree-$j$ "dual" forms $f\in D_j$ having given dimensions $H_u=H_{j-u}$ for its vector spaces of order u partials, for each u. The compressed case corresponds to the maximum dimensions for each $H_u$, hence, they are parametrized by an open dense subset of the projective space $P(D_j)$. More complicated socle types can be studied by compounding the Gorenstein case. Also, the article determines the dimensions of the families of nongraded compressed algebras. This paper is a dense one, not so easy to read, with a lot of content.
Several others, in particular Froberg and Laksov, wrote related articles particularly in the graded case.


12. Compressed algebras and components of the punctual Hilbert scheme, pp.146-185 in Algebraic Geometry, Sitges 1983, Lecture Notes in Math. vol 1124, Springer-Verlag 1985.


This article develops several further examples of "generic" Artin algebras, and summarizes what is known about the "topography" of the punctual Hilbert scheme $Hilb^n(P^n)$ when $r\ge 3$.


13. Tangent cone of a Gorenstein singularity, in Proceedings of the conference on Algebraic Geometry, Berlin 1985, H.Kurke and M. Roczen, eds., Teubnertexte zur math. vol. 92, Teubner, Leipzig, 1986, pp. 163-176.


After over 10 years of studying a mystery of how to approximate a GA algebra, I realized after looking at the differences of several columns of numbers, related to these "approximations" (and ignoring some calculation errors) that the Hilbert function of a non-graded GA algebra has a hidden symmetric structure, reported on here. This report presages the more definitive article [18] below, and includes some examples.


14. (with Joan Elias)The Hilbert function of a Cohen-Macaulay local algebra: extremal Gorenstein algebras, J. Algebra 110 (1987), 344-356.


The article studies higher dimensional analogues of the compressed and extremal Gorenstein Artin algebras, mainly in embedding codimension three. A preliminary result by L. Avramov and I, reported here, shows that there is a combined Hilbert function/ Loewy numbers invariant of a minimal reduction A, uniquely determined by the original higher dimensional local algebra B.


15. Hilbert scheme of points: Overview of last ten years, in Algebraic Geometry, Bowdoin 1985, S. Bloch, editor, PSPM #46 Part 2, Amer. Math. Soc.,Providence, 1987, pp. 297-320


A thorough survey, especially from the local point of view, omitting however the kinds of geometric problems involving special arrangements of points in $P^r$, that is well represented in the conference volume edited by Orecchia and Chiantini, somewhat later in 1994, and that I began to explore later.


16. The Hilbert function of a Gorenstein Artin algebra, p.348- 364 in Commutative Algebra, Proceedings of a Microprogram held June 15-July 2,1987 at MSRI, M. Hochster, C. Hunecke, J.D. Sally, Editors, Springer-Verlag, New York, 1989.


The main result is a proof that the associated graded algebra $A*=Gr_m(A)$ of a Gorenstein Artin local algebra has a stratification by subalgebras $A=C(0)\supset C(1)\cdots \supset C(s)$ whose quotients $Q(a)=C(a)/C(a+1)$. The stratification arises from the intersection of the Hilbert function and Loewy ($0:m^u$) strata on $A$, and is additional structure, beyond the algebra structure of $A*$. It follows that the Hilbert function of $A$ is the sum of shifted symmetric sequences. Thus, for example $H=(1,2,3,1,1)=(1,1,1,1,1)+(0,1,1,0)+(0,0,1,0,0)$ is the Hilbert function of a CI in two variables.


17. A nonunimodal graded Gorenstein Artin algebra in codimension five, with David Bernstein, 15 p., Comm. in Algebra 20(8), (1992), 2323-2336.


Using a construction of glueing an Artin algebra to its dual, we construct a sequence of families of GA algebras with non-unimodal Hilbert function, beginning with a compressed Artin algebra: the lowest embedding dimension of this series is 5. A similar construction had been adapted by Stanley to show that (1,13,12,13,1) is a Gorenstein sequence. Subsequently, Boij and Laksov reframed and formalized the class of examples, and the Boij showed that by repeating the process, he could obtain examples with arbitrarily many maxima, at the cost of high codimension.


18. Memoir:
    Associated graded algebra of a Gorenstein Artin algebra, 117 p.,AMS Memoir Vol. 107 #524 (1994).


Here I develop the ideas begun in [15] and [18], discussing the additional structure of the "Q-decomposition of H(A)" as an invariant of the local GA algebra A, and as an obstruction to certain deformations of A within the same Hilbert function (the family ZGOR(T)). The focus is on issues of parametrization, and of constructing GA algebras with a given Q-decomposition. A catalog is begun in low dimensions and lengths of the possible decompositions, and there are some results concerning the dimensions of the resulting families. The behavior of decompositions under linking is discussed. A strong Lefschetz theorem is shown for (general=nonhomogeneous) complete intersections in two variables. There is also a brief survey to that time of what is known about Gorenstein sequences, the Hilbert functions of graded GA algebras.


These topics strike me as a rich area, whose mining has just begun. One reason is that the focus of many geometers is on graded Gorenstein algebras where this approach yields no additional structure; and the classification of C^{inf} map germs is just nearing the lengths where this structure impinges.

19. (with Jacques Emsalem), Inverse system of a symbolic power I. J. Algebra, 174, (1995), 1080-1090


Determines the inverse system to a fat point (whose ideal is the power of a maximal ideal m_p at a point p of P^n), hence of the higher order neighborhoods of a variety. This was known to Terracini and apparently also to Lasker in the special case of order two, relevent to the Waring problem. A parallel study is made by Ehrenborg and Rota, also it appears in the more special case.


20. Inverse system of a symbolic power, II: the Waring problem for forms, J of Algebra, 174, (1995), 1091-1110.


This formally shows how the Waring problem for forms is a consequence of the Alexander-Hirschowitz interpolation theorem for order 2 vanishing on P^n. I also suggest a larger family of Waring problems, and indicate a connection with the existence of families Gor(T) of graded Gorenstein ideals having several irreducible components. These ideas were later further developed by Boij, Kanev and I in the book ([26],[28], and by Cho and I.


21. Inverse system of a symbolic power III: thin algebras and fat points, Compositio Math. 108, (1997), 319-356.


The article [22] implies that open questions about the Hilbert functions of generic sets of fat points on P^r -- the interpolation problem -- are equivalent to problems concerning the Hilbert function of ideals generated by certain powers of generic linear forms, via Macaulay duality. This article explicitly translates some conjectures of Froberg related to the latter problem into their analogues for the interpolation problem, and gives a "Main Conjecture" for the interpolation problem. It attempts to summarize what is known. The Froberg conjectured bounds do not impinge on the open problem of interpolation in P^2, but are important in P^3 and above.


22. Gorenstein Artin algebras, additive decompositions of forms and the punctual Hilbert scheme, in Commutative Algebra, Algebraic Geometry, and Computational Methods [Proceedings of Hanoi Conference in Commutative Algebra (1996)], D. Eisenbud, ed., Springer-Verlag (1999), 53--96. For downloadable dvi file: Gor-Art.tex.dvi


This both surveys results of several groups working on issues studied in [26], and attempts to present in brief form the main ideas of the approach V. Kanev and I took.


23. Book (With Vassil Kanev)
    Power Sums, Gorenstein Algebras, and Determinantal Loci, 345+xxvii p., to appear, Springer Lecture Notes in Mathematics #1721, December, 1999. More about the book


We completely rewrote and greatly extended the results of a 1996 preprint (see under Reports); we also added a substantial exposition, and attempted to make the book as user-friendly as we could. The subject had already changed as a consequence of the work of others who were able to use and in cases improve on the previous results. Appendix D compares the preprint and book results.


24. (With Steven L. Kleiman) The Gotzmann Theorems and the Hilbert Scheme, Appendix C to above book MS., 23p.
    More about the Appendix


An exposition of the Macaulay-Gotzmann results and their context (with some proofs indicated, but not that of the main Gotzmann result.) We also give a proof of a conjecture of D. Bayer concerning the equations of the Hilbert scheme Hilb^Q(P^r) (for a given polynomial Q). A final section gives some consequences for the study of Gor(T) for certain T containing a subsequence of maximal Macaulay growth.
25. (With Clare D'Cruz), High order vanishing ideals at $n+3$ points of $\mathbf P^n$, in Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998). J. Pure Appl. Algebra 152 (2000), no. 1-3, 75--82.



States some conjectures concerning an extreme case of the interpolation problem, gives some approaches and proves partial results.


26. (With Young Hyun Cho), Hilbert functions of Level Algebras, J. Algebra 241, (2001), 745--758. For downloadable dvi file of related preprint. Hilblev.dvi


After determining inequalities on the Hilbert functions of ideals, extending those of Macaulay, we determine upper and lower bounds on the Hilbert functions of level Artin algebras, those having socle in a single degree. We then consider level Artinian quotients of the coordinate ring of a zero-dimensional scheme $\Z\subset \mathbb P^n$, and give a natural upper bound for their Hilbert function, in terms of $H_\Z$ and the degree and type of $A$. We show that no sequence $H=(1,3,4,5,...,6,2)$ can occur as the Hilbert function of a level algebra: this shows that a main result of the previous paper concerning Artinian Gorenstein quotients of the coordinate ring of a Gorenstein punctual scheme, does not generalize simply to type 2 quotients: as there are smooth punctual schemes of Hilbert function $H_\Z=(1,3,4,5,6,6,\ldots ).


27. Dr. Ruth I. Michler's Research, p. 1-7 in ``Topics in Algebraic and Noncommutative Geometry: Proceedings in memory of Ruth Michler'', C. Melles, J. Brasselet, and G. Kennedy, editors, AMS Contemporary Mathematics series, # 324, 2003.


Summarizes Dr. Ruth I. Michler's research work, before her untimely death in an accident November 1, 2000 while visiting Northeastern as a NSF POWRE scholar.


28. (With Joachim Yameogo), The family G_T of graded Artinian quotients G_T of k[x,y] of given Hilbert function, Communications in Algebra, S. Kleiman volume, Vol. 31 \# 8, 3863--3916 (2003).

ArXiv alg-geom./9709021(The early version and posting of 1997 (see "Reports" iv. below) has been replaced with the 2003 preprint close to the extensively revised final article, main results are the same, both versions will be on the ArXiv)

Let $R=k[x,y]$ be the polynomial ring over an algebraically closed field $k$ of characteristic zero or of characteristic p>j. Let $T$ be a sequence of nonnegative integers that occurs as the Hilbert function of a length-$n$ Artinian quotient of $R$. The nonsingular projective variety $\G_T$ parametrizes all graded ideals $I$ of $R = k[x,y]$ for which the Hilbert function $H(R/I)=T$.
  We show that $\G_T$ is birational to a certain product $\SGrass(T)$ of small Grassmann varieties, and that over $k=\mathbb C$ the birational map induces an additive $\mathbb Z$ isomorphism of homology groups

\tau : H^{\ast}(\G_T) \longrightarrow H^{\ast}(\SGrass(T)),

  When mu is less than j, this isomorphism is not usually an isomorphism of rings. We determine the ring $H^*(\G_T)$ when
$T=T(\mu,j)=(1,2,\ldots ,\mu -1,\mu ,\ldots ,\mu,t_j=1)$
where $\G_T \subset \mathbb P^\mu\times \mathbb P^j$. In this case $\G_T$ is a desingularisation of the $\mu$-secant bundle $\Sec (\mu,j)$ of the degree-$j$ rational normal curve.
  We use this ring $H^*(\G_T)$ to determine the number of ideals satisfying an intersection of ramification conditions at different points. We also determine the classes in $H^*(\G_T)$ of the pullback of the singular locus of $\Sec (\mu ,j)$ and of the pullbacks of the higher singular loci.
  Let $E$ be a monomial ideal of $R$, satisfying $H(R/E)=T, \mid T\mid=n$: it corresponds to a partition $P(E)$ of $n$ having diagonal lengths T. A main tool is that the family of graded ideals having initial monomials $E$ is a cell $\mathbb{V}(E)$. We connect these cells to ramification conditions, using the Wronskian determinant, and to a ``hook code'' for $P(E)$.


29. Ancestor ideals of vector spaces of forms, and level algebras, J. Algebra 272 (2004), no. 2, 530--580.
ArXiv version: math.AC/030721

We study three algebras attached to a vector space of forms V\subset R_j, for R a polynomial ring: the algebra $R/(V)$, the level algebra $R/L(V))$, and the ``ancestor algebra'' R/(\overline{V}}$ combining these two. We determine dimension, codimension formulas for the Hilbert function strata in each case, and show that these strata satisfy a frontier property. A final application is to the simultaneous Waring problem for binary forms.


30. Betti strata of height two ideals (16 p.), J. Algebra 285 (2005) no. 2, 835--855.
ArXiv version: math.AC/0407364

We consider Betti strata of ideals in k[x,y] having given Hilbert function H=H(R/I): these strata are subschemes of the smooth projective variety $G(H)$ parametrizing all such quotients of Hilbert function H. We show a compact codimension formula for the Betti strata. When the field k is algebraically closed, we show that the strata are Cohen-Macaulay, and satisfy a frontier condition: the closure of a stratum is the union of itself and more special strata. Key tools throughout include properties of an invariant $\tau (V)$, which is the number of generators of the ancestor ideal $\overline{V}\subset R$, and previous results concerning the projective variety $\G(H)$. We also adapt a method of M.~Boij that reduces the calculation of the codimension of the Betti strata to showing that the most special stratum has the right codimension.


31. (with Hema Srinivasan), Some Gorenstein Artin algebras of embedding dimension four: Components of $PGOR(H)$ for $H=(1,4,7,\ldots ,1)$, Journal of Pure and Applied Algebra 201 (2005) # 1-3. 62--96.
    ArXiv version: math.AC/0412466


We study graded Gorenstein Artin (GA) algebras primarily of Hilbert functions $H=(1,4,7,\ldots )$. For $H=(1,4,7,h,7,4,1)$ we determina all possible graded Betti numbers of their minimal resolutions. We study these Betti strata geometrically, and determine irreducible components of the parameter variety PGOR(H), parametrizing all graded GA algebras of the given Hilbert function H. In most cases, we tie each Betti stratum to the geometry of points and curves in P^3, determined by the initial degree portions of the GA algebra, or by the net of quadrics given by I_2, if A = R/I.
We also show a structure theorem for GA algebras of Hilbert function $H=(1,4,7,\ldots ), where $I_2\cong wx, wy,wz\subset K[x,y,z,w]$. Finally we show that all Gorenstein sequences (sequences H that actually occur) beginning $H=(1,4,a,...)$ with $a\le 7$ satisfy the so called SI condition, that the first difference $\Delta H_{\le j/2}$ is an O-sequence (possible for an Artinian algebra). As a consequence of this result and the structure theorem, most PGOR(H), H = (1,4,7,b,...) have at least two irreducible components.


32. Hilbert functions of Gorenstein algebras associated to a pencil of forms, p. 273-286 in Proceedings of the Conference on Projective Varieties with Unexpected Properties, C. Ciliberto, A. Geramita, B. Harbourne, R. Mir\'o-Roig, K. Ranestad, editors (de Gruyter) (2005) ISBN-10:3-11-018160-6.
ArXiv version: math.AC/0412361

Consider a type two graded level algebra $A$, quotient of the polynomial ring $R=K[x_1,\ldots , x_r]$ in $r$ variables, defined by an inverse system $R\circ \langle F,G\rangle, F,G\in {\cal D}_j$ where ${\cal D} = K_{DP}[X_1,\ldots ,X_r]$ denotes the ring of divided powers, upon which $R$ acts by contraction. We have $A=R/\Ann (F,G)$. For each $\lambda\in K$, $F_\lambda = F+\lambda G$ determines a socle-degree $j$ Gorenstein algebra quotient of $A$,

  A_\lambda = R/\Ann (F_\lambda).

The family $F_\lambda$ is the pencil of degree-$j$ forms in $\mathcal D$ determined by the two-dimensional vector space $\langle F,G\rangle$. In our first main result we give bounds for the Hilbert function $H(A_\lambda)$, when $\lambda$ is generic, in terms of the Hilbert functions $H(R/\Ann (F))$ and $ H(R/\Ann (G))$ or in terms of the Hilbert function $H(A)$.
  We give several applications showing the impossibility of some candidate sequences to be Hilbert functions for type two level Artinian algebras. We also give an example of a compressed type two level Artinian algebra not having a compressed Gorenstein quotient of the same socle degree.
This result has been nicely generalized to type c level quotients of a type t level algebra by F. Zanello, in an article to appear in Proc. AMS,`` Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type'', see math.AC/0502466 for this.


33. (with R. Basili) Pairs of commuting nilpotent matrices, and Hilbert function, J. Algebra 320 # 3 (2008), 1235-1254. ``ArXiv AC: 0709.2304'' 18p.
    
    Consider associating to a partition P the generic Jordan partition Q(P) of a nXn nilpotent matrix commuting with a nilpotent Jordan matrix J_P of partition P. We first determine that the stable partitions P=Q(P) are those whose parts differe pairwise by at least two. We then show that in a pencil of nilpotent matrices A+\lambda J_P commuting with J_P, the generic element has partition that given by the partition P(H) associated to the graph of the Hilbert function $H[K[A,B]$, provided that dim_K K[A,B]=n.

Accepted/Submitted


34. (with M. Boij) Reducible family of height three level algebras, 20 p., to appear after revision, J. Algebra. ArXiv: math.AC/0707.2148.
    
    Determines families LevAlg(H) of height three level algebras, quotients of R=k[x,y,z], of types three and four having several irreducible components; two such families have liftings to points, although the component structure of one becomes more complex. Gives an infinite family of level height three Hilbert functions H(k) such that the number of irreducible components of LevAlg(H(k)) is bounded from below by a linear multiple of the socle degree.


35. (With Young Hyun Cho), Inverse systems of zero-dimensional schemes in $P^n$, preprint, July, 2000.
For downloadable dvi file: ChoI1.dvi;   pdf file: ChoI1.pdf

A study of the relation between the local inverse system of Macaulay for a local zero-dimensional scheme $Z$ on P^n, and the global inverse system for $I_Z$, the global defining ideal. Given generators of the local inverse systems -- corresponding to a spanning set of socle elements of the local algebras, we show that the global inverse system $L_\Z$ can be ``generated'' in a suitable sense by elements arising from homogenizations of the generators. Note that here $A=R/I_\Z$ is a dimension one ring, so we begin with an Artinian module(s), and determine from it a non-Artinian $R$-module $L_\Z$, with $R$ the coordinate ring of $\mathbb P^n$.
We then show that a local Gorenstein punctual scheme Z may be recovered from a sufficiently general Artinian Gorenstein quotient of $O_\Z$ -- equivalently, from a general element of $(I_Z)_j^{\perp}$ (the inverse system), provided j is large enough compared to the regularity and socle degree of Z. This allows in a sequel paper the construction of subfamilies of $PGOR(T)$ of Gorenstein Artin algebras of fixed Hilbert function $T$, fibred over a family of Gorenstein punctual schemes. The details turned out to be rather sensitive, and we wonder if there is a more functorial approach. As a byproduct, we show that the nonunimodal h-sequences \Delta H_Z studied by D. Bernstein and A. Iarrobino, M. Boij, and others, occur for local arithmetically Gorenstein punctual schemes that are cones. We also interpret the results in the language of generalized additive decompositions of forms $F$.

1. (with J. Emsalem) R$\'e$seaux de coniques et alg$`e$bres de longueur 7 associ$\'e$es (Nets of conics and length 7 algebras associated to them), 49p., U. of Paris VII, (1977).


Classifies nets of planar conics (three dimensional vector spaces of degree 2 forms in three variables), using a geometric approach. Later, C.T.C. Wall wrote a shorter work doing the classification less geometrically, but also in the real case.


2. Vanishing ideals at multiple points of $P^r$, p. 109-113, in Commutative Algebra, Extended abstracts of an International Conference at Vechta, 1994, W. Bruns, editor (1994).(BD 13).


In 1992 I had begun with Jacques Emsalem to extend the inverse systems approach that we were familiar with for local Artin algebras, to a global setting for P^r, projective r-space. Here I outlined some of the results obtained, that would be extended in the series of articles on Inverse systems I,II,III.


3. (With V. Kanev), The length of a homogeneous form, determinantal loci of catalecticants and Gorenstein algebras, preprint, 150 p.. 1996.


An early version of the book [23], this was circulated, and led to work by several other groups. This version was a rather more specialized monograph preprint than the actual book [23]. Since this preprint was circulated and referenced in some articles by others, we listed in Appendix E of [23] which sections of the book replace which sections of this preprint.


4. Graded ideals in k[x,y] and partitions,II: (with J. Yameogo) ramification and a generalization of Schubert calculus. 44p., preprint, U. Nice. (early version of #28)For Abstract and downloadable file: Alg-Geom #9709021


Note: The results appear in [33] below (in the S. Kleiman 2003 volume of Comm. in Algebra).

The main result is to use the hook code from [24] to show that the homology groups of G(T) (parametrizing graded ideals in k[x,y] of Hilbert function T) are the same as those of a "small" product of Grassmanians. A consequence is the study of fine stratifications of graded ideals in k[x,y] by the Weierstrass gaps at points of P^1. In certain, cases we determine the homology ring structure of (G(T)).

1. (with Joachim Yameogo) Graded ideals in $k[x,y]$, and Partitions,I: Partitions of diagonal lengths T and the hook code, preprint, 49 p.


Shows that a partition P is completely determined by T(P), the non-main diagonal lengths from its Ferrers graph, and by the distribution of the difference-one hooks, those whose arm lengths - leg lengths are one. The code is described in various ways; also the difference-a hooks are described as a combinatorial function of the difference-one hooks.




2. (With Young Hyun Cho), Gorenstein Artin algebras arising from punctual schemes in Projective Space, preprint, 2000. Revision in preparation.


First, we determine several global Hilbert functions $H_Z$ possible for (locally) Gorenstein schemes $Z$ in P^n, concentrated at a single point, and defined by a nongraded compressed Gorenstein Artin (GA) quotient of $\mathcal O_p$. As a consequence, we show many more examples of Gorenstein sequences $T=Sym(H_Z,j/2)$ for which Gor(T) has several irreducible components. These occur for either "small tangent space" or dimension reasons: one component corresponds to annihilating punctual schemes $Z$ that are as above; the second component corresponds to smooth annihilating schemes. The construction of the first component uses the results of the first joint article.


3. (With Hema Srinivasan), Some Gorenstein Artin algebras of embedding dimension four, II: Betti strata of $PGOR(H)$ for $H=(1,4,7,h,7,4,1)$, in preparation, 2002.


For $H=(1,4,7,h,7,4,1)$ we determine all possible graded Betti numbers of their minimal resolutions. We study these Betti strata geometrically, and determine irreducible components of the parameter variety PGOR(H), parametrizing all graded GA algebras of the given Hilbert function H. In most cases, we tie each Betti stratum to the geometry of points and curves in P^3, determined by the initial degree portions of the GA algebra, or by the net of quadrics given by I_2, if A = R/I.



4. Pencils of forms and Artinian level algebras.


Deals with families of Artinian algebras, related to article (39) on pencils. One goal of this preprint (not yet achieved) is to show the existence of families LEVALG(H) for embedding dimension r=3, type t=2, for which there are several irreducible components. A chief tool being the invariants described above, the Hilbert functions of Goresnstein quotients of $A$.


5. (with Franco Ghione, Gianni Sacchiero), Restricted Tangent Bundles of Rational Curves in $\mathbb P^r$, (revision of older preprint in progress)



6. Consecutive genera for irreducible Arithmetically Cohen-Macaulay Space Curves.


d Most recent in preparation:


7. (with R. Basili) An involution on N_B, the nilpotent commutator of a nilpotent Jordan matrix (pdf, 41 p., August, 2007).
   
   Introduces an involution that is a generalized transpose on the commutator C_B of B=J_P, that acts on the vertices of the digraph associated to a generic element of $U_B$, the maximal nilpotent subalgebra of C_B introduced by R. Basili.



8. (with M. Boij) Duality for central simple modules of Artinian Gorenstein algebras.


Aims to extend to nograded Gorenstein A, certain results of J. Watanabe and T. Harima concerning central simple modules of a graded Artinian Gorenstein algebra, using a generalization of results in "Associated Graded Algebras" to pairs of ideals.



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