Since the end of February 2023, the webpage of the AG Seminar has moved here!
This is the webpage of the Algebraic Geometry and Arithmetic Geometry Seminar.
The seminar features both research talks, as well as preprint talks presenting papers that appeared recently on arXiv in the algebraic geometry or the number theory sections and are of interest to the seminar participants.
If you wish to participate and or be added to the mailing list, contact me at s.marseglia (at) uu.nl.
In Block 1, the seminar takes place on Tuesdays at 11:15 in HFG409, followed by lunch, unless otherwise stated.
In Block 2, the seminar will take place on Wednesdays at 13:45 in HFG409, after lunch, unless otherwise stated.
Upcoming Talks
Block 1

September 28
(9:1510:15, BBG  223)
Annette HuberKlawitter (Freiburg)
Periods and ominimality
Abstract: (coming soon) (joint work with J. Commelin and P. Habbegger).

October 3
(11:1512:15, HFG 409)

October 10
(11:1512:15, HFG 409)
Shivang Jindal (Edinburgh)

October 17
(11:1512:15, HFG 409)

October 24
(11:1512:15, HFG 409)
Denis Nesterov (Vienna)

October 31
(11:1512:15, HFG 409)
Caleb Springer (University College London)

November 07
(11:1512:15, HFG 409)
Gerard van der Geer (UvABeijingLuxembourg)
Block 2

November 15
(13:4514:45, HFG 409)
[This week: Intercity Number Theory Seminar in Leiden]

November 22
(13:4514:45, HFG 409)
Yannik Schuler (Sheffield)
[This week also DIAMANT Symposium]

November 29
(13:4514:45, HFG 409)
Tudor Padurariu (Max Planck Bonn)

December 06
(13:4514:45, HFG 409)

December 13
(13:4514:45, HFG 409)
[This week: Intercity Number Theory Seminar in Amsterdam]

December 20
(13:4514:45, HFG 409)

December 27  no talk

January 03  no talk

January 10
(13:4514:45, HFG 409)

January 17
(13:4514:45, HFG 409)

January 24
(13:4514:45, HFG 409)

January 31
(13:4514:45, HFG 409)
Past Talks
2023

September 19
(11:1512:15, HFG 610)
Dusan Dragutinovic (Utrecht)
EkedahlOort types of stable curves
Abstract: In this talk, I will present some invariants of curves in positive characteristic p, such as the prank, the anumber, or the EkedahlOort type, and discuss intrinsic ways to define them. The main focus will be on Moonen's definition of EkedahlOort types of smooth curves in terms of HasseWitt triples. I will show that we can extend this definition to all stable curves. The description we obtain in this manner enables us to compute the dimensions of certain loci of curves. Finally, I will mention some new examples in characteristics p = 2 and p = 3.
[This week also: Intercity Number Theory Seminar in Utrecht]

September 12
(11:1512:15, HFG 610)
Carolina Tamborini (Utrecht)
Hodge theory and projective structures on compact Riemann surfaces
Abstract: A projective structure on a compact Riemann surface is an equivalence class of projective atlases, i.e. an equivalence class of coverings by holomorphic coordinate charts such that the transition functions are all Moebius transformations.
Any compact Riemann surface admits two canonical projective structures: one coming from uniformization's theorem, and one from Hodge theory. These yield two (different) families of projective structures over the moduli space Mg of compact Riemann surfaces. We wish to compare them and give a characterization of the Hodge theoretic family.

September 5
(11:1512:15, BBG 109)
Nicoló Piazzalunga (Rutgers)
4G Networks
Abstract: I'll introduce the equivariant Ktheoretic DonaldsonThomas theory for toric CalabiYau fourfolds, and construct its fourvalent vertex with generic plane partition asymptotics. Stringtheoretically, this is the count of BPS states of a system of D0D2D4D6D8branes in
the presence of a large NeveuSchwarz Bfield. The talk is based on 2306.12995, 2306.12405 and ongoing work.

June 28
(11:3012:30, HFG 409)
Sarah Arpin (Leiden)
Adding level structure to supersingular elliptic curve isogeny graphs
Abstract: The classical Deuring correspondence provides a roadmap between supersingular elliptic curves and the maximal orders which are isomorphic to their endomorphism rings. Building on this idea, we add the information of a cyclic subgroup of prime order N to supersingular elliptic curves, and prove a generalisation of the Deuring correspondence for these objects. We also study the resulting ellisogeny graphs supersingular elliptic curve with levelN structure, and the corresponding graphs in the realm of quaternion algebras. The structure of the supersingular elliptic curve ellisogeny graph underlies the security of a new cryptographic signature protocol, SQISign, which is proposed to be resistant against both classical and quantum attack.

June 23 (14:0015:00, HFG 409)
Aline Zanardini (Leiden University)
Pencils of plane cubics revisited
Abstract: In recent joint work with M. Hattori we have considered the problem of classifying linear systems of hypersurfaces (of a fixed degree) in some projective space up to projective equivalence via geometric invariant theory (GIT). And we have obtained a complete and explicit stability criterion. In this talk I will explain how this criterion can be used to recover Miranda's description of the GIT stability of pencils of plane cubics.

June 21
(11:3012:30, BBG 165)
Lou van den Dries
(University of Illinois at UrbanaChampaign)
Transseries and Hardy Fields
Abstract: Transseries are formal series involving exp and log. The differential field of transseries is a universal domain for algebraic differential equations with asymptotic side conditions (in analogy with the field of complex numbers being a universal domain for algebraic geometry). I will then apply this to Hardy fields. Until recently there were only very limited ways of extending Hardy fields, but recently we have achieved a more or less complete overview. I will start from scratch and define/explain transseries and Hardy fields, including examples.
(Joint work with Matthias Aschenbrenner and Joris van der Hoeven.)

June 14
(11:3012:30, HFG 409)
Gergely Berczi (Aarhus)
Tautological intersection theory of Hilbert scheme of points
Abstract: While the Hilbert scheme of points on surfaces is pretty wellunderstood, the Hilbert scheme over manifolds presents a mixture of pathological and unknown behaviour: our knowledge of their components, singularities and deformation theory is very limited.
After a brief survey we report on a new approach to calculate tautological intersection numbers of geometric subsets which play crucial role in enumerative geometry applications. We present a ChernSegretype positivity conjecture for tautological integrals coming from global singularity theory.

June 7 :
(11:3012:30, HFG 409)
Jeongseok Oh (Imperial College, KIAS)
Koszul factorisations of moduli spaces and invariants
Abstract: Invariants on moduli spaces are defined to be integrations over the virtual fundamental classes or Euler characteristics of the virtual structure sheaves. Roughly a virtual structure sheaf is a Koszul resolution of the structure sheaf and the virtual fundamental class is its Chern character (multiplied by Todd class).
In fact their first algebraic constructions by LiTian and BehrendFantechi look different. Here we explain these two are the same in a reasonable circumstance. Furthermore, we explain the use of Koszul factorisations rather than Koszul resolutions gives rise to KiemLi localisations of LiTian and BehrendFactechi constructions. These are joint works with Bumsig Kim and Bhamidi Sreedhar.
We can use these to produce invariants of moduli spaces of stable sheaves on CalabiYau 4folds as explained in the joint work with Richard Thomas.

May 24 (11:3012:30, HFG 409)
Nancy Abdallah (University of Borås)
Lefschetz properties of Artin Gorenstein algebras in low codimensions
Abstract: The weak and strong Lefschetz properties (WLP and SLP) has been much studied for Artinian algebras. Codimension two Artinian algebras over a field of characteristic zero have the strong Lefschetz property. It is open whether Artin Gorenstein (AG) algebras of codimension c=3 satisfy the weak or strong Lefschetz properties. Harima, Migliore, Nagel, and Watanabe proved that complete intersection algebras of codimension three satisfy the weak Lefschetz property. For c=4, Gondim showed that WLP always holds for regularity r <= 4 and gives a family where WLP fails for any r >= 7, building on an earlier example of Ikeda of failure for r=5.
In this talk we go through an overview of Lefschetz properties over AG algebras and we give some recent results in codimensions 3 and 4. In codimension 4 we explore relations between Lefschetz properties of an AG algebra A and the free resolutions of A. Lefschetz properties of the Milnor algebras of projective hypersurfaces will be discussed.

May 17 (11:3012:30, HFG 409)
Samir Canning (ETH)
Semitautological systems and the cohomology of the moduli space of curves
Abstract: I will introduce the notion of semitautological systems, which are systems of subalgebras with minimal set of functoriality properties of the cohomology rings of the moduli spaces of stable curves. They are designed to study the structure of the cohomology of the moduli spaces of stable curves beyond the tautological ring. I will give a criterion for a given semitautological system to span all of cohomology in a given degree. Using this criterion and other results about the moduli space of curves, both topological and algebrogeometric, I will give several applications. These applications include a complete description of the thirteenth cohomology of the moduli space of stable n pointed curves of genus g for all g,n and that all cohomology classes of sufficiently high degree are tautological. This is joint work in progress with Hannah Larson and Sam Payne.

May 10, (11:3012:30 MIN 2.01) Thorsten Schimannek (CNRS LPTHE)
Counting curves on nonKaehler CalabiYau 3folds with Topological Strings
Abstract: In general, a projective CalabiYau threefold with nodal singularities does not admit a Kaehler small resolution.
This happens in particular if the exceptional curves are torsion in homology.
In this talk we will discuss how the classical relationship between topological string theory, enumerative geometry and mirror symmetry generalizes to this setting.
After recalling some of the ideas from the smooth case, we will both explain the physical intuition behind the generalization and translate it into a concrete mathematical proposal.
At the end of the talk, if time permits, we will highlight some open questions that follow from this proposal, related to DonaldsonThomas theory, noncommutative geometry, and FJRWtheory.

May 3, (13.3014.30, HFG 409) ChiaFu Yu (Academia Sinica, Taiwan)
Arithmetic invariants on supersingular EkedahlOort (EO) strata for g=4
Abstract: The supersingular locus is one of main interests in algebraic geometry in characteristic p, and can be described in terms of polarised flag type quotients (PFTQs) in the sense of Li and Oort. The description for g=3 is rather explicit and is exploited by Karemaker and Yobuko and myself for investigating the arithmetic invariants of supersingular abelian threefolds, namely, the endomorphism rings and automorphism groups of them, confirming Oort's conjecture for g=3. In this talk we shall explain a general method for investigating the arithmetic invariants on supersingular EO strata, and report the progress of the joint work with Karemaker for g=4.

April 26, (13:3015:30, HFG 409) Minicourse by Carolina Tamborini (UU)
Families of Galois covers of the line: examples and construction

April 25, (13:3015:30, HFG 610) Minicourse by Carolina Tamborini (UU)
Some results on totally geodesic subvarieties in the Torelli locus

April 20, (13:3015:30, HFG 610) Minicourse by Carolina Tamborini (UU)
Torelli map, its differential, its second fundamental form, and the Lie bracket map

April 19, (13:3014:30, HFG409) research talk by Oliver Leigh (Uppsala)
The Blowup Formula for the Instanton Part of VafaWitten Invariants on Projective Surfaces
Abstract: In this talk I will present a blowup formula for the generating series of virtual χ_ygenera for moduli spaces of sheaves on projective surfaces. The formula is related to a conjectured formula for topological χ_ygenera of Göttsche, and is a refinement of a formula of VafaWitten relating to Sduality. I will also discuss the proof of the formula, which is based on the blowup algorithm of NakajimaYoshioka for framed sheaves on ℙ^2. This talk is based on joint work with Nikolas Kuhn and Yuuji Tanaka.

April 18, (13:3015:30, HFG 610) Minicourse by Carolina Tamborini (UU)
Siegel space as symmetric space

April 5, (13:30, HFG409) research talk by Fabian Reede (Hannover)
Line bundles on noncommutative algebraic surfaces
Abstract: Let X be a complex algebraic surface. Let A be a (noncommutative) coherent O_Xalgebra whose generic stalk is a central division algebra. Then the pair (X,A) can be thought of as a noncommutative algebraic surface. This talk deals with the moduli space of line bundles on (X,A), i.e., of locally free Amodules of rank one, in the cases where X has Kodaira dimension zero or is Fano. It turns out that the moduli space can be compactified by adding torsionfree Amodules of rank one. We will study e.g. the smoothness of this compactification and the deformation theory of the sheaves involved. This is partly joint work with U. Stuhler and with N.Hoffmann.

March 22, (13:30, HFG409) research talk by Sara Mehidi (Bordeaux)
Extending torsors via log schemes
Abstract: We give here an approach of the problem of extending torsors defined on the generic fiber of a family of curves. The question is to extend each of the structural group and the total space of the torsor above the family. This problem has been studied by many researchers, starting by the first ideas of Grothendieck who solved the case of a constant group of order prime to the residual characteristic. When we are interested in algebraic varieties from an arithmetic point of view, it is natural to consider torsors under a finite flat group that is not necessarily constant: we talk about fppf torsors. In fact, we know from the literature that there are cases where the problem does not have a solution in this setting.
So the idea is to look for a solution in a larger category, namely the category of logarithmic torsors. We will show in particular that the existence of such an extension amounts to extending group functors and morphisms between them. Then, we will compute the obstruction for the extended log torsor to lift into an fppf one. Finally, we give an example of a computation of an extension of a torsor over a given hyperelliptic curve as an application of our results.

March 15, (13:30, HFG409) research talk by Henry Liu (Oxford):
Multiplicative vertex algebras and wallcrossing in equivariant Ktheory
Abstract: I will give an overview of recent progress in wallcrossing
for equivariant Ktheoretic invariants of moduli of sheaves, based on
a new framework of Joyce. A multiplicative version of vertex algebras
plays a central role in this and related stories. I will give some
applications to refined VafaWitten theory, the 3fold DT/PT
correspondence, and, if time permits, also some speculation about
modularity and Sduality.
 March 1, (13:30, HFG409) research talk by Stefano Marseglia (UU): CohenMacaulay type of endomorphism rings of abelian varieties over finite fields
Abstract: In this talk, we will speak about the (CohenMacaulay) type of the endomorphism ring of abelian varieties over a finite field with commutative endomorphism algebra.
We will exhibit a condition on the type of End(A) implying that A cannot be isomorphic to its dual. In particular, such an A cannot be principally polarised or a Jacobian. This is partly joint work with Caleb Springer.
 February 15, (13:30, BBG069) research talk by Andreas Braun (Durham): Hodge classes on CalabiYau fourfolds
Abstract: A crucial question in string theory concerns its set of solutions,
particularly those giving rise to an effective fourdimensional
description. The most general such solutions are found by specifying a
CalabiYau fourfold equipped with an elliptic fibration, together with a
Hodge class.
Without assuming any background I will briefly explain
how these objects arise in physics, and which constraints they need to
obey. After formulating some of the central questions and conjectures
that arise in this context, I will discuss some recent progress.
 February 8, (13:30, HFG409) research talk by Hyeonjun Park (KIAS): Cosection localization via derived algebraic geometry
Abstract: Cosection localization is one of the most powerful tools in virtual enumerative geometry. In this talk, we revisit cosection localization from the perspective of derived algebraic geometry. I will explain derived reduction by (1)shifted closed 1forms and localization through homotopical intersection theory. I will also provide an intrinsic description of cosectionlocalized virtual cycles using (2)shifted symplectic structures. This is based on joint works with Younghan Bae and Martijn Kool, with Dhyan Aranha, Adeel Khan, Alexei Latyntsev, and Charanya Ravi, and with YoungHoon Kiem.
 February 1, (11:00, HFG409) research talk by Olivier de Gaay Fortman (Hannover): Real moduli spaces, unitary Shimura varieties and nonarithmetic lattices
Abstract: Hodge theory can sometimes be used to identify a moduli space of complex varieties with a complex ball quotient, or an open subset of such a space. I will explain that similar things happen for moduli of real varieties. Real moduli spaces of smooth varieties are often not connected, however  to get a connected moduli space one is led to allow some singularities. It turns out that, very similar to the way in which the connected components of the space of smooth varieties embed into the larger moduli space, real arithmetic ball quotients can be glued together to form a large real ball quotient. Unitary Shimura varieties provide the right framework for this glueing procedure. I will explain how this works, constructing nonarithmetic lattices in PO(n,1) for every n.
 January 18, (10:00, HFG409) research talk by Mick van Vliet (UU): Tame geometry and Hodge theory
Abstract: Tame geometry, made precise by the concept of ominimal structures, has recently led to some interesting developments in algebraic geometry. In the first half of this talk I will motivate and explain the definition of ominimal structures, and review some remarkable theorems that hold in the resulting framework of tame geometry. In the second half of the talk, based on work of Bakker, Klingler, and Tsimerman (1810.04801), I will give an overview of a recent application of tame geometry to Hodge theory.
 January 11, (11:00, HFG409) research talk by Lars Halvard Halle (University of Bologna): Degenerations of Hilbert schemes and relative VGIT
Abstract: This talk will be a report on joint work with K. Hulek and Z. Zhang. First
I will explain how some central results in VGIT can be extended to a
relative setting. After this, I will discuss an application of relative
VGIT to the study of certain degenerations of Hilbert schemes of points.
2022
 December 7, (HFG409) research talk by Leo Herr (Leiden University): The Rhizomic Topology
Abstract: What is a sheaf on a log scheme X? If we take the ordinary etale topology, we ignore the log structure. Taking the log étale topology, even the structure "sheaf" O_X is not a sheaf! The same goes for M_X, \overline M_X. We introduce a new "rhizomic" topology on log schemes coarser than the log etale topology. Will this be enough?
 November 30, (HFG409) research talk by Navid Nabijou (QMU London): Roots and logs in the enumerative forest
Abstract: Logarithmic and orbifold structures provide two different paths to the enumeration of curves with fixed tangencies to a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify birational invariance as the key property distinguishing the two theories. The logarithmic theory is stable under strata blowups of the target, while the orbifold theory is not. By identifying a suitable system of “slopesensitive” blowups, we define a “limit" orbifold theory and prove that it coincides with the logarithmic theory. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is muchbetter understood.
 November 23, (HFG409) research talk by Mar Curco Iranzo (UU): Generalised Jacobians of modular curves and their Qrational torsion
Abstract: The Jacobian J0(N) of the modular curve X0(N) has received much attention within arithmetic geometry for its relation with cusp forms and elliptic curves. In particular, the group of Qrational points on X0(N) controls the cyclic Nisogenies of elliptic curves. A conjecture of Ogg predicted that, for N prime, the torsion of this group comes all from the cusps. The statement was proved by Mazur and later generalised to arbitrary level N into what we call generalised Ogg’s conjecture.
Consider now the generalised Jacobian J0(N)m with respect to a modulus m. This algebraic group also seems to be related to the arithmetic of X0(N) through the theory of modular forms. In the talk we will present new results that compute the Qrational torsion of J0(N) for N an odd integer with respect to a cuspidal modulus m. These generalise previous results of Yamazaki, Yang and Wei. In doing so, we will also discuss how our results relate to generalised Ogg’s conjecture.
 November 16, (HFG409) research talk by Francesca Carocci (EPFL): BPS invariant from non Archimedean integrals
Abstract: We consider moduli spaces of onedimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes.
Working over a nonarchimedean local field F, we define a natural measure on the Fpoints of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on the moduli spaces with respect to this measure is independent of the Euler characteristic.
Analogous statements hold for (meromorphic or not) Higgs bundles.
Recent results of MaulikShen and KinjoCoseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles.
This is a joint work with Giulio Orecchia and Dimitri Wyss.
 October 26, (HFG409) research talk by Reinier Schmiermann: On Classifying Continuous Constraint Satisfaction Problems
Abstract: The computational complexity class of the existential theory of the reals contains problems which can be reduced to checking whether a system of polynomial equations has a solution over the real numbers. The complexity of a lot of problems in computational geometry turns out to be captured by this class (they are complete for this class). These completeness proofs often use the completeness of a specific Continuous Constraint Satisfaction Problem (CCSP) as an intermediate step. We attempt to give a more systematic analysis of the computational complexity of these CCSPs, and show that a large class of CCSPs is complete for the existential theory of the reals.
In this talk, I will give a introduction to computational complexity, the existential theory of the reals, and CCSPs. Then I will state our results, and give a sketch of the proof. This talk is based on joint work with Tillmann Miltzow.
 October 19, (HFG409) Carel Faber, preprint talk on "On the Chow and cohomology rings of moduli spaces of stable curves" by Canning and Larson, arXiv:2208.02357
 October 12 (HFG409), research talk by Remy van Dobben de Bruyn: A variety that cannot be dominated by one that lifts
Abstract: The recent proofs of the Tate conjecture for K3 surfaces over finite fields start by lifting the surface to characteristic 0. Serre showed in the sixties that not every variety can be lifted, but the question whether every motive lifts to characteristic 0 is open. We give a negative answer to a geometric version of this question, by constructing a smooth projective variety that cannot be dominated by a smooth projective variety that lifts to characteristic 0.
 October 5, research talk by Dusan Dragutinovic:
Computing binary curves of genus five
Abstract. In this talk, we will present algorithms used to determine, up to isomorphism over $\F_2$, all genus five curves defined over $\F_2$ (together with the sizes of their $\F_2$automorphism groups). Furthermore, we will discuss the outcome considering the Newton polygons of computed curves and mention the obtained stack count $\mathcal{M}_5(\F_2)$.
 September 21, Boaz Moerman, preprint talk on "Weak approximation and the Hilbert property for Campana points" by Nakahara and Streeter, arXiv:2010.12555
 July 6 (11:00  12:00, HFG611), research talk by Valentijn Karemaker
 June 28 (11:00  13:00, HFG611), research talk by Pol van Hoften
 June 22 (11:15, HFG610) Carolina Tamborini, "Punctual characterization of the unitary flat bundle of weight 1 PVHS and application to families of curves" by GonzálezAlonso and Torelli, arXiv:2101.03153
 June 8 (HFG610), Wilberd van der Kallen, "Frobenius Splittings", arXiv:1208.3100
 May 31 (HFG610), Dirk van Bree, "When are two HKR isomorphisms equal?" by Huang, arXiv:2205.04439
 May 25 (HGF610), Marta Pieropan, "Heights on stacks and a generalized BatyrevManinMalle conjecture" by Ellenberg, Satriano and ZureickBrown, arXiv:2106.11340
 May 9 (Duistermaat), Marta Pieropan, "Global Frobenius liftability I & II" by Achinger, Witaszek and Zdanowicz, arXiv:1708.03777 and arXiv:2102.02788
 March 7, Sebastián Carrillo Santana, "Values of zetaone functions at positive even integers" by Kobayashi and Sasaki, arXiv:2202.11835
 Feb. 28, Carolina Tamborini, "The ColemanOort conjecture: reduction to three key cases" by Moonen, arXiv:2201.11971
 Feb. 14, Boaz Moerman, "Tamagawa measures on universal torsors and points of bounded height on Fano varieties" by Salberger, article,
and "Compter des points d'une variété torique" by de la Bretèche, article
 Feb. 7, Sergej Monavari, "On the motive of the Quot scheme of finite quotients of a locally free sheaf" by Ricolfi, arXiv:1907.08123
 Jan. 24, Dirk van Bree, "Using the internal language of toposes in algebraic geometry" by Blechschmidt, arXiv:2111.03685
2021
 Nov. 29 (HFG611), Reinier Schmiermann, "Components and singularities of Quot schemes and varieties of commuting matrices" by Jelisiejew and Šivic, arXiv:2106.13137
 Nov. 22 (KBGAtlas), Dusan Dragutinovic, "The existence of supersingular curves of genus 4 in arbitrary characteristic" by Kudo, Harashita and Senda, arXiv:1903.08095
 Nov. 8, Dirk van Bree, "Unramified division algebras do not always contain Azumaya maximal orders" by Antieau and Williams, arXiv:1209.2216
 Nov. 1, Stefano Marseglia, "On matrices of endomorphisms of abelian varieties" by Zarhin, arXiv:2002.00290, and "Lattices in Tate modules" by Poonen and Rybakov, arXiv:2107.06363
 Oct. 11, Carel Faber, "A nonhyperelliptic curve with torsion Ceresa class" by Beauville, arXiv:2105.07160, and "A nonhyperelliptic curve with torsion Ceresa cycle modulo algebraic equivalence" by Beauville and Schoen, arXiv:2106.08390
 Oct. 4, Marta Pieropan, "Sums of four squareful numbers" by Shute, arXiv:2104.06966