Schedule for: 16w5114 - Lefschetz Properties and Artinian Algebras

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Many important invariants of ideals in a polynomial ring can be read off from the locations of the zeros in their corresponding Betti table, for example the regularity, projective dimension, etc. We consider two problems on Betti tables of monomial ideals generated in degree 2 (edge ideals, after polarization): 1. Strand connectivity: can rows in the Betti table have internal zeros? [Conca, Weildon] 2. Subadditivity: for ti the maximal j for which "i,j is nonzero, must ta+b <= ta + tb? [Herzog-Srinivasan, Avramov-Conca-Iyengar] We show that for the first question the answer is NO for the first 2 rows and YES otherwise. We use it in showing that for the second question the answer is YES for b = 1, 2, 3 (for b = 1 this was proved by Herzog-Srinivasan, for all monomial ideals). Via Hochster formula, our proofs are topological-combinatorial.

Problems

Problems

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

The Jordan type of a nilpotent matrix is the partition giving the sizes of the Jordan blocks in the normal Jordan form of the matrix. In this talk we discuss all partitions that have a fixed partition Q as the generic Jordan type in their nilpotent commutator. We report on a joint work with A. Iarrobino, B. Van Steirteghem and R. Zhao in which we provide a complete description of ball such partitions for a partition Q with at most two parts. In particular we arrange all such partitions in a table that we denote by T (Q). We then report on an ongoing joint project with M. Boij, A. Iarrobino, B. Van Steirteghem and R. Zhao in which we study the equations of loci in T (Q).

We give a characterization, depending on the characteristic p > 0 of the field k, for the values d1, ...,dn (under certain triangle-inequality-like restrictions) for the quotient of a polynomial ring in n variables by the powers of the variables raised to powers d1, ...,dn to have WLP.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In a joint work with Rosa Maria Mir`o-Roig and Giorgio Ottaviani (Canad. J. Math. 65, 2013), we established a closed relationship, due to apolarity, between homogeneous Artinian ideals I of the polynomial ring which fail the Weak Lefschetz Property - WLP - and projective varieties X satisfying at least one Laplace equationof order s2, i.e. such that all the s-osculating spaces have dimension strictly less than expected. Thanks to this connection, we were able to classify the smooth (and quasi smooth) toric rational threefolds parametrized by cubics, satisfying a Laplace equation of order 2, extending a classical theorem of E. Togliatti for surfaces. Equivalently, we classified the monomial Artinian ideals of cubics in 4 variables. We also formulated a conjecture to extend this result to ideals generated by cubic monomials in any dimension. The conjecture has been successively proved by Rosa Maria Mir-Roig and Mateusz Michalek (arXiv 1310.2529). The assumption that the variety is toric allows to exploit the combinatorial methods, studying the associated polytope. More recently in a joint work with Rosa Maria Mir`o-Roig (arXiv 1506.05914), we have started to investigate the same problems for Artinian ideals of the polynomial ring generated by monomials of any degree d in any number of variables and failing the WLP. These are also called Togliatti systems. Since the picture becomes soon much more involved than in the case of cubics, we have restricted our attention mainly to Togliatti systems that are minimal and smooth, adressing the question of their minimal and maximal number of generators. We have solved this question, and also classified the systems with minimal number of generators, or number of generators close to the minimal, and found new classes of examples. After shortly recalling the relationship mentioned above, I would like to speak of the more recent results on Togliatti systems of any degree d

Mezzetti, Mir´o-Roig, and Ottaviani showed that in some cases the failure of the weak Lefschetz property can be used to produce a variety satisfying a Laplace equation. We define a graded algebra to have a Lefschetz defect of ! in degree d if the rank of the multiplication map of a general linear form between the degree d−1 and degree d components has rank ! less than the expected rank. Mezzetti and Mir´o-Roig recently explored the minimal and maximal number of generators of ideals that fail to have the weak Lefschetz property, i.e., to have a positive Lefschetz defect. In contrast to this, we will discuss constructions of ideals that have large Lefschetz defects and thus can be used to produce toric varieties satisfying many Laplace equations.

Problems

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We introduce a family of standard bigraded binomial Artinian Gorenstein algebras, whose combinatoric structure characterizes the ones presented by quadrics. These algebras provide, for all socle degree grater than two and in sufficiently large codimension with respect to the socle degree, counter-examples to Migliore-Nagel conjectures. One of them predicted that Artinian Gorenstein algebras presented by quadrics should satisfy the weak Lefschetz property. We also prove a generalization of a Hessian criterion for the Lefschetz properties given by Watanabe, which is our main tool to control the Weak Lefschetz property.

The Bernstein-Gelfand-Gelfand correspondence is an equivalence between certain derived categories over the polynomial algebra S and the exterior algebra E. In down to earth terms, it allows one to define a functor from S-modules to complexes of free E-modules with linear differential, and vice versa. This connects to the multiplication map used to investigate Lefschetz properties, but it seems little has been done to explore this. Some specific classes of objects that might be amenable to study using these techniques are (Artinian reduction of) squarefree monomial ideals, and ideals generated by products of linear forms

Problems with U.Nagel and H.Schenck, with invitation to others to work on these problems.

Watanabe’s bold conjecture states that every Artinian complete intersection algebra (generated in degree one) can be embedded in an lArtinian complete intersection cut out by quadratic forms. We verify this conjecture for coinvariant rings of (finite) complex reflection groups generated by involutory reflections, which includes all finite Coxeter 3 groups. For a Coxeter group associated to a flag variety, the quadratic complete intersection algebras that we construct correspond to the cohomology rings of certain ”resolutions” of the flag variety, due to Bott and Samelson. These so-called Bott-Samelson algebras have been studied extensively by Soergel, whose work eventually led Elias and Williamson to a purely algebraic proof of the notorious Kazhdan-Lusztig positivity conjecture. Along the way, I will try to highlight some of these remarkable results of Soergel and Elias-Williamson, and their surprising connection with the strong Lefschetz property. (Joint with Larry Smith)

Let A be a graded complete intersection over a field and B the monomial complete intersection with generators of the same degrees as for A. The EGH conjecture says that if I is a graded ideal in A, then there should be an ideal J in B such that B/J and A/I have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property. In the course of proof we introduce a new definition which we call the “matching property for graded Gorenstein algebras. Then we prove that the matching property implies the Sperner property. This part is an independent result of the EGH conjecture.

Problems

Fix a ground field F and denote by R = F[V ] the graded algebra of homogeneous polynomial functions on V = Fn. Write Rg for the subalgebra of R pointwise fixed by an element g 2 GLn(F). For g1, g2, ..., gk 2 GLn(F), following Bott and Samelson, we introduce the algebra BS(g1, g2, ..., gk) = R ⌦Rg1 R ⌦Rg2 · · ·R ⌦Rgk R which we call the Bott-Samelson algebra and an artinian reduction BS(g1, g2, ..., gk) = F ⌦R BS(g1, g2, ..., gk) of it which we call the reduced Bott-Samelson algebra. We prove that for a k -tuple of reflections s1, s2, ..., sk the reduced Bott- Samelson algebra is a complete intersection algebra and, if F has characteristic zero, then BS(s1, s2, ..., sk) has the strong Lefschetz property. If ⇢ : G ! GLn(F) is a faithful reflection representation of a finite group G we show how to construct an embedding of the coinvariant algebra F[V ]G into a reduced Bott-Samelson algebra BS(s1, s2, ..., sk) where s1, s2, ..., sk are reflections generating G, provided that the coinvariant algebra is fixed point free, meaning that the fixed point set of the group G acting on its coinvariant algebra consists of the scalar multiples of the identity element alone. In the nonmodular case, i.e., the case where the order |G| of G is prime to the characteristic of the ground field, coinvariant algebras are always fixed point free, we construct such a Bott-Samelson embedding F[V ]G ,! BS(s1, s2, ..., sk) which is a degree one map between Poincar´e duality algebras. In certain favorable cases, e.g., if all the reflections in G have order 2, which is the case for all but three of the primitive complex reflection groups of degree at least 3, as was proven by H.F. Blichfeldt, we deduce by the Subring Theorem, that F[V ]G has the strong Lefschetz property provided the field F has characteristic zero. For the three exceptions we have a game plan to show that they too have coinvariant algebras with the SLP, and it is this that I would like to talk about.

In a recent paper, Mezzetti, Miro-Roig and Ottaviani [Mezzetti et al., “Laplace equations and the weak Lefschetz property”, Canad. J. Math. 65 (2013) 634654] highlight the link between rational varieties satisfying a Laplace equation and artinian ideals failing the Weak Lefschetz Property. In a joint work with Roberta di Gennaro and Giovanna Ilardi [“Singular hypersurfaces characterizing the Lefschetz properties”, J. London Math. Soc. (2014) 89 (1):194-212] we extended their link to the more general situation of artinian ideals failing the Strong Lefschetz Property. We characterize the failure of the SLP (which includes WLP) by the existence of special singular hypersurfaces (cones for WLP). Thank to this characterization we relate, for instance, the splitting type of derivation bundles associated to a line arrangements to the failure of the SLP of some artinians ideals

We discuss connections between Lefschetz properties and the study of Hilbert functions of (fat) points as well as the theory of line arrangements. To this end, we begin by considering a finite set Z of points in the plane with the property that, for some integer j, the dimension of the linear system of plane curves of degree j + 1 through the points of Z and having multiplicity j at a general point is unexpectedly large. We give criteria for the occurrence of such unexpected curves and describe the range of their degrees. Inspired by work of Di Gennaro, Ilardi, and Vall`es, we relate properties of Z to properties of the arrangement of lines dual to the points of Z. In particular, we get a new interpretation of the splitting type of a line arrangement, and we show that the existence of an unexpected curve is equivalent to the failure of a certain Lefschetz property. This implies a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements. This is based on joint work with D. Cook II, B. Harbourne, and J. Migliore

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.