# Schedule for: 16w5114 - Lefschetz Properties and Artinian Algebras

Beginning on Sunday, March 13 and ending Friday March 18, 2016

All times in Banff, Alberta time, MST (UTC-7).

Sunday, March 13 | |
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16:00 - 17:30 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre) |

17:30 - 19:30 |
Dinner ↓ 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. (Vistas Dining Room) |

20:00 - 22:00 | Informal gathering (Corbett Hall Lounge (CH 2110)) |

Monday, March 14 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

08:45 - 09:00 | Introduction and Welcome by BIRS Station Manager (TCPL 201) |

09:00 - 09:50 | Mats Boij: Betti tables of ideals and monomial ideals (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
Eran Nevo: On vanishing patterns in Betti tables of edge ideals ↓ 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. (TCPL 201) |

11:10 - 12:00 | Problem Sessions: short talks from participants (TCPL 201) |

11:10 - 11:30 |
Hal Schenck: Problems - WLP for various combinatorial rings ↓ Problems (TCPL 201) |

11:30 - 11:45 |
Martina Juhnke-Kubitzke: Problems - SLP on simplicial 3-polytopes ↓ Problems (TCPL 201) |

11:45 - 12:00 | Satoshi Murai: Problems - SLP on simplicial 3-polytopes, continued (TCPL 201) |

12:00 - 13:30 | Lunch (Vistas Dining Room) |

13:00 - 14:00 |
Guided Tour of The Banff Centre ↓ Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus. (Corbett Hall Lounge (CH 2110)) |

13:45 - 15:00 | Work in groups (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:00 | Work in groups (TCPL 201) |

16:10 - 16:40 | Discussion - Reports of groups (TCPL 201) |

16:50 - 17:20 |
Leila Khatami: Equations of loci in tables of commuting Jordan types ↓ 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). (TCPL 201) |

17:30 - 18:00 |
Adela Vraciu: WLP for monomial complete intersections in positive characteristic ↓ 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. (TCPL 201) |

18:00 - 19:30 |
Dinner ↓ 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. (Vistas Dining Room) |

Tuesday, March 15 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:50 |
Emilia Mezzetti: Togliatti systems and Artinian ideals failing weak Lefschetz ↓ 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 (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
David Cook II: Large Lefschetz defects ↓ 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. (TCPL 201) |

11:10 - 12:00 | Problem Sessions: short talks from participants (TCPL 201) |

11:10 - 11:40 |
Anthony Iarrobino: Problems - Jordan type partitions determined by multiplication by nilpotent elements ↓ Problems (TCPL 201) |

11:40 - 12:00 | Problems - further, from participants (TCPL 201) |

12:00 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 13:40 |
Group Photo ↓ 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! (TCPL Foyer) |

13:40 - 15:00 | Work in groups (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:00 | Work in groups (TCPL 201) |

16:10 - 16:40 | Discussion - Reports of groups (TCPL 201) |

16:50 - 17:20 |
Rodrigo José Gondim Neves: Lefschetz properties for Artinian Gorenstein algebras presented by quadrics ↓ 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. (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Wednesday, March 16 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:50 |
Hal Schenck: Bernstein Gelfand-Gelfand correspondence and WLP ↓ 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 (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 | Problem Sessions: short talks from participants (TCPL 201) |

10:30 - 11:00 |
Juan Migliore: Connections between WLP and geometry,including Hartshorne-Rao modules (with U. Nagel and H. Schenck) ↓ Problems with U.Nagel and H.Schenck, with invitation to others to work on these problems. (TCPL 201) |

11:00 - 12:00 | Work in groups (TCPL 201) |

12:00 - 12:30 | Discussion - Reports of groups (TCPL 201) |

12:30 - 14:00 | Lunch (Vistas Dining Room) |

13:30 - 17:30 | Free Afternoon (Banff National Park) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Thursday, March 17 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:50 |
Christopher McDaniel: Bott-Samelson algebras and Watanabe’s bold conjecture ↓ 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) (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
Junzo Watanabe: The EGH conjecture and the Sperner property of complete intersections ↓ 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. (TCPL 201) |

11:10 - 12:00 | Problem Sessions: short talks from participants (TCPL 201) |

11:10 - 11:30 |
Rodrigo José Gondim Neves: Problems - On higher Hessian and the Lefschetz properties ↓ Problems (TCPL 201) |

11:30 - 11:50 | Junzo Watanabe: Problems - On higher Hessian and the Lefschetz properties (TCPL 201) |

11:50 - 12:10 | Other problems from participants welcome (TCPL 201) |

12:00 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 15:00 | Work in groups (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:00 | Work in groups (TCPL 201) |

16:10 - 16:40 | Discussion - Reports of groups (TCPL 201) |

16:50 - 17:20 |
Larry Smith: The Strong Lefschetz Property for Coinvariant Algebras of Reflection Groups ↓ 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. (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Friday, March 18 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 09:30 |
Jean Valles: Singular hypersurfaces characterizing the Lefschetz properties ↓ 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 (TCPL 201) |

09:40 - 10:10 |
Uwe Nagel: Unexpected curves, line arrangements, and Lefschetz properties ↓ 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 (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:50 - 12:00 | Reports of Groups and further discussions (TCPL 202) |

11:30 - 12:00 |
Checkout by Noon ↓ 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. (Front Desk - Professional Development Centre) |

12:00 - 13:30 | Lunch from 11:30 to 13:30 (Vistas Dining Room) |