Meeting of the Swedish, Spanish and Catalan Mathematical Societies
Umeå, June 12 – June 15, 2017
Special session: Algebraic Geometry and Commutative Algebra
Preliminary Program
Tuesday June 13
Time
Speaker
Talk title
14.00–14.30 Kaie Kubjas
Maximum likelihood geometry for group-based phylogenetic models
14.30–15.00 Stiofáin Fordham On the height of the formal group associated to a smooth projective
hypersurface
15.00–15.30 Laura Costa
Ulrich bundles on ruled surfaces
15.30–16.00
COFFEE BREAK
16.00–16.30 Jens Forsgård
Singularities of Discriminants
16.30–17.00 Luis Narváez
Hasse-Schmidt derivations versus classical derivations
17.00–17.30 Alberto Castaño Irregular Hodge filtration of hypergeometric D-modules
1730–18.00 Adson Banda
Characterization of Half Exact Coherent Functors over Principal
Ideal Domains and Dedekind domains
Wednesday June 14
Time
14.00–14.30
14.30–15.00
15.00–15.30
15.30–16.00
16.00–16.30
16.30–17.00
Speaker
Ignacio Garcı́a-Marco
Roser Homs
Ferran Dachs-Cadefau
Talk title
Noether resolutions in dimension 2
On low Gorenstein colength of Artin local rings
Computing jumping numbers in higher dimensions
COFFEE BREAK
Dolors Herbera
A construction of big Maximal Cohen Macaulay modules
Josep Álvarez Montaner D-modules, Bernstein-Sato polynomials, and F-invariants of
direct summands
17.00–17.30 Leif Melkersson
Finiteness properties of local cohomology modules
1730–18.00 Santiago Zarzuela
Spectral sequences in local cohomology
Thursday June 15
Time
14.00–14.30
14.30–15.00
15.00–15.30
15.30–16.00
Speaker
Alessandro De Stefani
Joan-Carles Naranjo
Vı́ctor González Alonso
Talk title
Globalizing F-invariants
On the Xiao conjecture for families of plane curves
Entropy of automorphisms of supersingular K3 surfaces
COFFEE BREAK
Abstracts
D–modules, Bernstein–Sato polynomials, and F –invariants
of direct summands
Josep Àlvarez Montaner
Universitat Politècnica de Catalunya
In this work we study D-module structures over a ring that is a direct summand of
the polynomial or the formal power series ring with coefficients over a field. We prove
that localizations and local cohomology modules have finite length and we will show
the existence of a Bernstein-Sato polynomial in this non-regular framework. Time
permiting, we will turn our attention to some invariants of singularities in positive
characteristic. We prove the discreteness and rationality of F-jumping numbers and
we also extend some relations between F-thresholds and roots of the Bernstein-Sato
polynomial.
This is joint work with C. Huneke and L. Núñez-Betancourt [ÀMHNB].
Characterization of Half Exact Coherent Functors over
Principal Ideal Domains and Dedekind domains
Adson Banda
Linköpings Universitet
We give necessary and sufficient conditions for a coherent functor over a principal
ideal domain (PID) and over a Dedekind domain to be half exact. We show that
every half exact coherent functor over a PID and more generally over a Dedekind
domain arises from a complex of projective modules.
Under the assumption that A is noetherian commutative ring, we consider the
exact sequence
/ F0
/0
/F
(1)
F (A) ⊗ − α
and show that αM : F (A) ⊗ M → F (M ) is injective for any A–module M with
projective dimension at most 1. We then show that if F is a half exact coherent
functor over a PID, then F0 in (1) is left exact. In this case F0 ∼
= HomA (N, −) for
some finitely generated module N . We further show that over a PID, the sequence
(1) splits, that is, F ∼
= (F (A) ⊗ −) ⊕ F0 . This shows that F arises from a complex
since both F (A) ⊗ − and Hom(N, −) arise from a complex of projective modules.
1
Irregular Hodge filtration of hypergeometric D-modules
Alberto Castaño Domı́nguez
Technische Universität Chemnitz
Mixed Hodge modules provide the correct framework to incorporate singularities,
which arise usually in geometry, to the differential equations induced by variations
of Hodge structures. Nevertheless, they are not the most general context since
those singularities are always regular. Sabbah and Mochizuki have constructed an
impressive extension of this setting, becoming the so-called theory of mixed twistor
modules, allowing to introduce irregular singularities.
Recently, Sabbah (based on joint work with Esnault and Yu and ideas of Deligne
and Katzarkov, Kontsevich and Pantev) introduced in [Sab] the so-called irregular
Hodge filtration, allowing one to assign numerical invariants (i.e., irregular Hodge
numbers) to certain mixed twistor modules. He has shown that all rigid irreducible
D-modules on the projective line admit a unique irregular Hodge filtration provided
that the formal local monodromies are unitary. Rigid D-modules are particularly
interesting since they can be algorithmically constructed from simple objects by an
algorithm due to Arinkin and Katz. Among the first and best understood examples
of such rigid D-modules are the classical hypergeometric D-modules. In the regular
case, Fedorov has recently given in [Fed] a closed formula for the Hodge numbers
using the work of Dettweiler and Sabbah.
In this talk, we will report on how one can determine irregular Hodge numbers for
a general irregular hypergeometric D-module. For a specific class formed by those
modules with a purely irregular singularity, a direct calculation is possible and yields
numbers related to other Hodge theoretic invariants such as Hodge spectra of nondegenerate Laurent polynomials. If time permits, we will also explain the possible
strategies to achieve the general case.
The content of this talk is based on joint work [CDS] with Christian Sevenheck.
Ulrich bundles on ruled surfaces
Laura Costa
Universitat de Barcelona
An Ulrich bundle on a smooth projective variety is a vector bundle that admits a
completely linear resolution as a sheaf on the projective space. They appeared in
commutative algebra, being associated to maximal Cohen Macaulay graded modules
with maximal number of generators. In my talk, I’ll focus the attention on the
existence of special rank two Ulrich bundles on ruled surfaces. This is based on
joint work with Marian Aprodu and Rosa Maria Miró–Roig [ACMR].
2
Computing jumping numbers in higher dimensions
Ferran Dachs–Cadefau
Martin-Luther-Universität Halle-Wittenberg
Multiplier ideals and jumping numbers are invariants that encode relevant information about the structure of the ideal to which they are associated. A first part of
this talk will be devoted to introduce some basics about multiplier ideals in the case
of 2-dimensional local rings.
In the second part of the talk, we will introduce some results for the multiplier ideals in the higher-dimensional case. For this, we introduce the notion of
π-antieffective divisors, a generalization of antinef divisors to higher dimensions.
Using these divisors, we present a way to find a small subset of the classical candidate jumping numbers of an ideal, containing all the jumping numbers. Moreover,
many of these numbers are automatically jumping numbers, and in many other
cases, it can be easily checked.
The presented results are part of a joint work with Hans Baumers [BDC].
Globalizing F-invariants
Alessandro De Stefani
KTH Royal Institute of Technology
The Hilbert-Kunz multiplicity and the F-signature are two important numerical
invariants, defined for local rings of prime characteristic. They are subtly connected
with the theory of singularities, and they often provide a good measure of how
ill-behaved a ring can be. We will survey some classical results on the HilbertKunz multiplicity, and we will discuss how to extend this notion to rings that are
not necessarily local, in a way that still detects relevant information. This gives a
possible way to meaningfully extend these concepts to more geometric objects, such
as algebraic varieties. Time permitting, we will discuss analogous results for the
F-signature. The talk is based on joint work with Thomas Polstra and Yongwei Yao
[DSPY].
On the height of the formal group associated to a smooth
projective hypersurface
Stiofáin Fordham
University College Dublin
Given a smooth projective hypersurface over a finite field, we can associate to the
defining polynomial a descending chain of ideals after Àlvarez-Montaner–Blickle–
Lyubeznik [AMBL05]. Then, a result of Boix–De Stefani–Vanzo [BDSV15] related
the point at which the chain stabilizes to the height of the associated formal group
in the case of an elliptic curve. I will describe the interpretation of their result using
Frobenius splitting methods, and work in progress in extending their result to the
case of Calabi-Yau hypersurfaces.
3
Singularities of Discriminants
Jens Forsgård
Texas A&M University
We will discuss singularities of A-discriminants in terms of the Horn–Kapranov
uniformization. Applications to the study of dual defect toric varieties will be given.
This is joint work with J. Maurice Rojas.
Noether resolutions in dimension 2
Ignacio Garcı́a-Marco
Aix-Marseille Université
Let R := K[x1 , . . . , xn ] be a polynomial ring over an infinite field K, and let I ⊂
R be a homogeneous ideal with respect to a weight vector ω = (ω1 , . . . , ωn ) ∈
(Z+ )n such that dim(R/I) = d. In this work we study the minimal graded free
resolution of R/I as A-module, that we call the Noether resolution of R/I, whenever
A := K[xn−d+1 , . . . , xn ] is a Noether normalization of R/I. When d = 2 and I
is saturated, we give an algorithm for obtaining this resolution that involves the
computation of a minimal Gröbner basis of I with respect to the weighted degree
reverse lexicographic order. In the particular case when R/I is a 2-dimensional
semigroup ring, we also describe the multigraded version of this resolution in terms
of the underlying semigroup. Whenever we have the Noether resolution of R/I or
its multigraded version, we obtain formulas for the corresponding Hilbert series of
R/I, and when I is homogeneous, we obtain a formula for the Castelnuovo-Mumford
regularity of R/I.
As an application of the results for 2-dimensional semigroup rings, we provide a
new upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of a
projective monomial curve. Finally, we describe the multigraded Noether resolution
and the Macaulayfication of either the coordinate ring of a projective monomial
curve C ⊆ PnK associated to an arithmetic sequence or the coordinate ring of any
n−1
.
canonical projection πr (C) of C to PK
All the results that will be presented in this talk are based on joint work with
Isabel Bermejo, Eva Garcı́a Llorente, and Marcel Morales [BGLGMM].
4
Entropy of automorphisms of supersingular K3 surfaces
Vı́ctor González–Alonso
Leibniz Universität Hannover
It is known that the topological entropy of an automorphism of an algebraic surface
is either zero or the logarithm of a Salem number, but not much is known about
which Salem numbers can actually be realized. In this talk I will present a joint
work with Simon Brandhorst [BGA] in the case of supersingular K3 surfaces, giving a
method to construct an automorphism with given entropy and showing in particular
which minimal Salem numbers occur in this case.
A construction of big Maximal Cohen Macaulay modules
Dolors Herbera
Universitat Autònoma de Barcelona
A module is pure projective if it is isomorphic to a direct summand of a direct sum
of finitely presented modules. There is a big amount of work done on the study of
finitely generated (hence, finitely presented) modules over commutative noetherian
rings, but not so much is known on direct summands of infinite direct sums of such
modules. With our work we want to give some insight on this problem.
Let R be a local commutative noetherian ring. If R is complete then all pure
projective modules are direct sums of finitely generated modules, but we give plenty
of examples showing that this is far from true in the non-complete case. However
there is still a close relation between pure-projective R-modules and pure projective
R̂-modules, as we can prove that two pure-projective modules P and Q are isomorphic as R-modules if and only if P ⊗R R̂ ∼
= Q ⊗R R̂ as R̂-modules. The proof of
such result is modeled on Přı́hoda’s one that two arbitrary projective modules are
isomorphic if and only if they are isomorphic modulo the Jacobson radical [P0̌7].
A rather difficult question is to determine which pure-projective R̂-modules are
extended from R-modules. We will present an answer to this problem when R is
a one dimensional domain and for direct summands of an arbitrary direct sum of
copies of a single finitely generated R-module. Our techniques to do that use heavily
the results in [HP10] and one of our main source of examples is [Wie01].
The results that will be presented are part of an ongoing long joint project with
Pavel Přı́hoda and Roger Wiegand.
5
On low Gorenstein colength of Artin local rings
Roser Homs
Universitat de Barcelona
I will introduce the notion of Gorenstein colength of an Artin local k-algebras and
how to compute it. I will give a complete characterization of k-algebras of colengths
0, 1 and 2 in terms of its Macaulay inverse system and discuss the problem for higher
colengths. This is a joint work with Joan Elias [EH].
Maximum likelihood geometry for group-based phylogenetic
models
Kaie Kubjas
Aalto–yliopisto
Based on Matsen’s work on inequalities for group-based phylogenetic models in
Fourier coordinates, we study polynomial equations and inequalities that cut out
group-based models in original coordinates. We apply this knowledge to the study
of boundaries and maximum likelihood estimation on group-based models. In particular, we use the degree of an algebraic variety and numerical algebraic geometry
to obtain the maximum likelihood estimate exactly for small group-based models.
This talk is based on joint work with Dimitra Kosta [KK].
Finiteness properties of local cohomology modules
Leif Melkersson
Linköpings Universitet
I will give a survey of finiteness properties of local cohomology modules and discuss
some topics related to this as finiteness of Ext and Tor-modules. I will also discuss
some open problems in the area. If time permits I will prove some new results of
mine.
6
On the Xiao conjecture for families of plane curves
Joan–Carles Naranjo
Universitat de Barcelona
Xiao’s conjecture deals with the relation between the natural invariants present on
a fibred surface f : S → B: the irregularity q of S, the genus b of the base curve B
and of the genus g of the fibre of f . In a paper with M.A. Barja and V. GonzálezAlonso [BGAN] we have proved the inequality q − b ≤ g − c, where c is the Clifford
index of the generic fibre. This gives in particular a proof of the (modified) Xiao’s
conjecture, q − b ≤ g/2 + 1, for fibrations whose general fibres have maximal Clifford
index. In this talk we we will report on recent progress on the Xiao’s conjecture for
fibrations whose generic fibre is a plane curve. More precisely we will focus on a joint
work with F. Favale and G. P. Pirola [FNP] where we have proved the conjecture
for quintic plane curves, we also have proved the inequality q − b ≤ g − c − 1 for
plane cuves of any degree greater than or equal to 5.
Hasse-Schmidt derivations versus classical derivations
Luis Narváez Macarro
Universidad de Sevilla
In this talk we will review on the notion of Hasse–Schmidt derivation of a commutative algebra, and we will see how these natural objects allow us to define good
differential smoothness properties and to describe rings of differential operators in
any characteristic (equal or unequal), and modules over them. On the other hand,
the notion of Hasse–Schmidt derivation gives rise to a notion of integrability for
(usual) derivations. This property always hold in characteristic zero. So, one can
expect that integrability should play a role in the different behavior of singularities
in characteristic zero and in positive characteristic.
The content of this talk is based on [NM12], [NM09], [FLNM05] and [FLNM03].
7
Spectral sequences in local cohomology
Santiago Zarzuela
Universitat de Barcelona
Spectral sequences are often used to compute local cohomology functors. In this
talk I’m going to review how to use them in order to calculate local cohomology
from the primary decomposition of an ideal I in a commutative Noetherian ring
R. In the homological case, we shall deal with the computation of several generalized local cohomology functors supported on I. In the cohomological case we shall
mainly be concerned with the computation of the local cohomology of R = I. The
construction of these spectral sequences is done by means of the computation of the
left and right derived functors of the direct and inverse limits in terms of the homology (or cohomology) of a particular explicit complex, that we call homological (or
cohomological) Roos complex. In each case, one can also give sufficient conditions
in order to guarantee the degeneration of the corresponding spectral sequence. As
a guiding cases we have in mind the results obtained by Àlvarez-Garcı́a- Zarzuela
[ÀMGLZA03] and G. Lyubeznik [Lyu07] in the homological case.
The content of this talk is based on a joint work in progress with Josep ÀlvarezMontaner and Alberto F. Boix [ÀMBZ].
8