On the degree two entry of a Gorenstein hh-vector and a conjecture of Stanley

In this note we establish a (non-trivial) lower bound on the degree two entry $h_2$ of a Gorenstein $h$-vector of any given socle degree $e$ and any codimension $r$. In particular, when $e=4$, that is for Gorenstein $h$-vectors of the form $h=(1,r,h_2,r,1)$, our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say $f(r)$, that $h_2$ may assume. In fact, we show that $$\lim_{r\to \infty} {f(r)\over r^{2/3}}= 6^{2/3}.$$ In general, we wonder whether our lower bound is sharp for all integers $e\geq 4$ and $r\geq 2$.Comment: A few minor changes. To appear in Proc. of the AM

Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree $i+1$ entry of a Gorenstein $h$-vector, in terms of its entry in degree $i$. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given $r$ and $i$, all Gorenstein $h$-vectors of codimension $r$ and socle degree $e\geq e_0=e_0(r,i)$ (this function being explicitly computed) are unimodal up to degree $i+1$. This immediately gives a new proof of a theorem of Stanley that all Gorenstein $h$-vectors in codimension three are unimodal. Our second main theorem is an asymptotic formula for the least value that the $i$-th entry of a Gorenstein $h$-vector may assume, in terms of codimension, $r$, and socle degree, $e$. This theorem broadly generalizes a recent result of ours, where we proved a conjecture of Stanley predicting that asymptotic value in the specific case $e=4$ and $i=2$, as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree $i= \lfloor \frac{e}{2} \rfloor$.Comment: Several minor changes; to appear in J. Algebr

A characterization of Gorenstein Hilbert functions in codimension four with small initial degree

The main goal of this paper is to characterize the Hilbert functions of all (artinian) codimension 4 Gorenstein algebras that have at least two independent relations of degree four. This includes all codimension 4 Gorenstein algebras whose initial relation is of degree at most 3. Our result shows that those Hilbert functions are exactly the so-called {\em SI-sequences} starting with (1,4,h_2,h_3,...), where $h_4 \leq 33$. In particular, these Hilbert functions are all unimodal. We also establish a more general unimodality result, which relies on the values of the Hilbert function not being too big, but is independent of the initial degree.Comment: A few changes. Final version, to appear in Math. Res. Let

Minimal Resolution of Relatively Compressed Level Algebras

A relatively compressed algebra with given socle degrees is an Artinian quotient $A$ of a given graded algebra R/\fc, whose Hilbert function is maximal among such quotients with the given socle degrees. For us \fc is usually a ``general'' complete intersection and we usually require that $A$ be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that finding this value is equivalent to the Fr\"oberg Conjecture. We then turn to the minimal free resolution of a level algebra relatively compressed with respect to a general complete intersection. When the algebra is Gorenstein of even socle degree we give the precise resolution. When it is of odd socle degree we give good bounds on the graded Betti numbers. We also relate this case to the Minimal Resolution Conjecture of Mustata for points on a projective variety. Finding the graded Betti numbers is essentially equivalent to determining to what extent there can be redundant summands (i.e. ``ghost terms'') in the minimal free resolution, i.e. when copies of the same $R(-t)$ can occur in two consecutive free modules. This is easy to arrange using Koszul syzygies; we show that it can also occur in more surprising situations that are not Koszul. Using the equivalence to the Fr\"oberg Conjecture, we show that in a polynomial ring where that conjecture holds (e.g. in three variables), the possible non-Koszul ghost terms are extremely limited. Finally, we use the connection to the Fr\"oberg Conjecture, as well as the calculation of the minimal free resolution for relatively compressed Gorenstein algebras, to find the minimal free resolution of general Artinian almost complete intersections in many new cases. This greatly extends previous work of the first two authors.Comment: 31 page

On the componentwise linearity and the minimal free resolution of a tetrahedral curve

A tetrahedral curve is an unmixed, usually non-reduced, one-dimensional subscheme of projective 3-space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines. The second and third authors have shown that these curves have very nice combinatorial properties, and they have made a careful study of the even liaison classes of these curves. We build on this work by showing that they are "almost always" componentwise linear, i.e. their homogeneous ideals have the property that for any d, the degree d component of the ideal generates a new ideal whose minimal free resolution is linear. The one type of exception is clearly spelled out and studied as well. The main technique is a careful study of the way that basic double linkage behaves on tetrahedral curves, and the connection to the tetrahedral curves that are minimal in their even liaison classes. With this preparation, we also describe the minimal free resolution of a tetrahedral curve, and in particular we show that in any fixed even liaison class there are only finitely many tetrahedral curves with linear resolution. Finally, we begin the study of the generic initial ideal (gin) of a tetrahedral curve. We produce the gin for arithmetically Cohen-Macaulay tetrahedral curves and for minimal arithmetically Buchsbaum tetrahedral curves, and we show how to obtain it for any non-minimal tetrahedral curve in terms of the gin of the minimal curve in that even liaison class.Comment: 31 pages; v2 has very minor changes: fixed typos, added Remark 4.2 and char. zero hypothesis to 5.2, and reworded 5.5. To appear, J. Algebr