# Seminar Algebraische Geometrie

**Summer Semester 2021**

The talks will be held on Mondays at 16:15 via Webex (https://www.webex.com).

Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m557416b71c55a1d50a22cc33627f21b3

Meeting number: 121 099 5804

Password: Toric*****Variety without stars.

The schedule of the talks, with abstracts, appears at https://researchseminars.org/seminar/algebraic_geometry_FU

The schedule of the talks, without abstracts, is below.

12.04.2021 Simon Felten (Universität Mainz)

The logarithmic Bogomolov-Tian-Todorov theorem

19.04.2021 Mattia Talpo (University of Pisa)

26.04.2021

03.05.2021 Jonathan Lai (Imperial College London)

10.05.2021 No talk - you can watch a talk at the online seminar "Algebra, Geometry, and Combinatorics" https://sites.google.com/view/agconline/

17.05.2021 Lara Bossinger (UNAM Oaxaca)

24.05.2021 No talk - Pfingstmontag

31.05.2021 Marian Aprodu (University of Bucharest)

07.06.2021

14.06.2021

21.06.2021

28.06.2021

05.07.2021

12.07.2021

**Winter Semester 2020/2021**

**Schedule:**

**02.11.2020 kein Seminar**

09.11.2020 Karin Schaller (FU-Berlin)

** Stringy Chern classes**

**16.11.2020 Amelie Flatt (HU-Berlin)**

**Displaying Ext**

**23.11.2020 Noah Giessing (Potsdam)**

**Localization in algebraic K-theory**

**30.11.2020 Noah Giessing (Potsdam)**

**Rational homogeneous spaces**

07.12.2020 TALK POSTPONED!

**Alvaro Liendo (Universidad de Talca, Chile)**

**14.12.2020**

**Lena Walter (FU-Berlin)**

**Toric Newton-Okounkov functions**

We initiate a combinatorial study of Newton-Okounkov functions on toric varieties with an eye on the rationality of asymptotic invariants of line bundles. In the course of our efforts we identify a combinatorial condition which ensures a controlled behavior of the appropriate Newton-Okounkov function on a toric surface. Our approach yields the rationality of many Seshadri constants that have not been settled before. This is joint work with Christian Haase and Alex Kueronya.

**04.01.2021**

**Livia Campo (Nottingham)**

**Sarkisov links for index 1 Fano 3-folds**** in codimension 4**

As possible outcomes of the Minimal Model Program, Mori fibre spaces play a crucial role in Birational Geometry, and is therefore important to study how they connect to each other. The Sarkisov Program provides tools to analyse birational transformations between Mori fibre spaces. In this talk I will examine the case of codimension 4 Fano 3-folds with Fano index 1 and explain how machineries likegraded rings and GIT can help to study their birational classes. This study also leads to some conclusions regarding their Picard rank.

webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525

Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41

**11. 01.2021 **

**Joaquin Moraga (Princeton)**

**The Jordan property for local fundamental groups**

For a finite subgroup G of GL_n(C), we can find a normal abelian subgroup A<G so that its index in G is bounded by a constant c(n), which only depends on n. In this talk, we prove that a similar statement holds for the local fundamental group of n-dimensional klt singularities. Then, we show applications of this statement to the study of iteration of Cox rings of Fano type varieties.

webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525)

Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41

**18.01.2021 PLEASE NOTE THAT THE DATE OF THIS TALK IS NOW 15.02.2021:**

**Vladimir Lazić (Saarbruecken)**

** Which properties of the canonical class depend only on its first Chern class**

**25.01.2021 Due to illness this talk will take place on March 1st.**

**Valentina Kiritchenko (Moscow)**

**Push-pull operators and Newton-Okounkov polytopes**

**01.02.2021 **

**Hendrik Suess (Manchester)**

**TBA**

webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525)

Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41

**08.02.2021**

**Alvaro Liendo (Talca, Chile)**

**Characterization of algebraic varieties by their groups of symmetries**

webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525)

Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41

**15.02.2021**

**Vladimir Lazić (Saarbruecken)**

** Which properties of the canonical class depend only on its first Chern class?**

Given a projective variety X with mild singularities and a line bundle L on it, it is a natural question to determine which properties of L are encoded by its first Chern class. I will argue that most of the interesting properties of the canonical bundle of X, such as its effectivity or semiampleness, are indeed almost always encoded by its first Chern class. The results are a consequence of the Minimal Model Program and Hodge theory, and are new even on surfaces. This is joint work with Thomas Peternell.

webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525)

Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41

**22.02.2021**

**Michel Brion (Grenoble)**

**Homomorphisms of algebraic groups: representability and rigidity**

The talk will discuss the following questions.

Given two algebraic groups G and H (over the complex numbers simplicity), is there a "moduli space" M for the homomorphisms from G to H? In the affirmative, how to describe the H-orbits on M, where H acts by conjugation of homomorphisms? By a result of Grothendieck, M exists when G is diagonalizable and H is linear; then every H-orbit in M is open. On the other hand, M does not exist when G is the additive group and H is the multiplicative group. We will sketch how to extend Grothendieck's results to the case where G is reductive (and H arbitrary).

webEx; Meeting number: 121 255 9136, Meeting password: JcgRQneN525)

Meeting link: https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=m27d010244c4481dca71d7e5fe9d4ac41

**01.03.2021**

**Valentina Kiritchenko (Moscow)**

**Push-pull operators and Newton-Okounkov polytopes**

A classical result of Schubert calculus is an inductive description of Schubert cycles using divided difference (or push-pull) operators in Chow rings of projective bundles. We define convex geometric analogs of push-pull operators by replacing Chow rings with polytope rings (Khovanskii-Pukhlikov rings). We apply push-pull operators to describe Newton-Okounkov polytopes of Schubert varieties and of Bott-Samelson varieties. In particular, Grossberg-Karshon cubes and Minkowski sums of Fegin-Fourier-Littelmann-Vinberg polytopes can be constructed inductively using convex grometric push-pull operators. All definitions will be given in the talk.

New Link!

https://fu-berlin.webex.com/fu-berlin-en/j.php?MTID=mc13de3b6c3fa3fb9d708d2e6e8bc2d25https

Meeting number: 121 021 0842 Password: APvpnWM4N82

**Sommer Semester 2020**

**Schedule**

**20.04.2020**

**Johannes Hofscheier (Nottingham) **(Webex talk--details below)

**A geometric proof of the generalised Mukai conjecture for horospherical Fano varieties**

**Abstract**: Horospherical varieties naturally generalise toric and flag varieties. They form a rich class of algebraic varieties admitting an action by a reductive group with an open dense orbit. In this talk, I will present joint work in progress with Giuliano Gagliardi on a geometric proof of the generalised Mukai conjecture for horospherical Fanovarieties. Our approach uses a geometric characterisation of topic varieties using log pairs which was conjectured by Shokurov and recently proven byBrown-McKernan-Svaldi-Zong.

Meeting-Link: https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898).

**27.04.2020**

**Elena Martinengo (Turin) **(Webex-talk--details below)

**On degeneracy loci of equivariant bi-vector fields on a smooth toric variety.**

On a smooth toric variety $X$ of dimension $n$, we study equivariant bi-vector fields, i.e. global sections of the second symmetric power of the homolorphic tangent bundle that are equivariant with respect to the toric action. We are interested in the degeneracy loci, that are the loci in which therank of such a bi-vector field is less or equal some integer $k$. In particular, in the spirit of a Bondal conjecture, we prove that the locus where the rank of an equivariant bi-vector field is $\leq 2k$ is not empty and has at least a component of dimension $\geq 2k+1$, for all integers $k > 0$ such that $2k < n$. The same is true also for $k = 0$, if the toric variety is smooth and compact. While for the non compact case, the locus in question has to be assumed to be non empty.

Meeting-Link:

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

**18.05.2020**

**Rostislav Devyatov (Bonn)** (Webex talk--details below)

Meeting-Link:

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

**25.05.2020**

**Andreas Hochenegger (Pisa) **(Webex talk--details below)

**Ulrich bundles on projective bundles**

Abstract: Given some smooth projective variety X, it is a difficult question whether X admits an Ulrich bundle, which are locally free sheaves fulfilling strong conditions on its cohomology. In this talk I will present some constructions of Ulrich bundles in the case that X itself is a projective bundle. This is about a joint work in progress with Klaus Altmann.

Meeting-Link:

https://fu-berlin.webex.com/fu-berlin/j.php?MTID=m7e8fe38c0f6e7005a499ba5f085feee5 (Meeting number 849 418 444, password wTbiMZCb898)

**01.06.2020**

**No talk! Pentecost holiday.**

**08.06.2020**

**Lutz Hille (Muenster) **** **(Webex talk--details below)

**Inverse limits of moduli of quiver representations, Chow quotients and Losev-Manin moduli spaces**

**Abstract: **Motivated by the construction of the compactifications of the moduli space of marked curves of genus zero with n marks (often called Mumford-Knudsen moduli space) we consider a similar construction for certain quiver representations. This construction uses King's construction of the moduli space of \Theta-stable quiverrepresentations and a certain inverse limit over all possible weights \Theta.

In a special case we recover the Mumford-Knudsen, the Losev-Manin and all the Hassett compactifications for a certain choice of a quiver, a dimension vector and a certain set of Weights.

In this talk I consider the toric situation, that started as joint work with Altmann about 25 years ago and is still a challenge from a combinatorial point of view. In a particular situation we get the Losev-Manin compactification, however, using quivers, it is obvious how to generalize this approach (joint work with Mark Blume).

Meeting-Link:

** **

**15.06.2020**

**Jan Stovicek (Prague)** (Webex talk--details below)

**Pure-injective quasi-coherent sheaves on elliptic curves**

**Abstract:** The aim of the talk is to explain a joint contribution with Alessandro Rapato to the classification of indecomposable pure-injective quasi-coherent sheaves over the elliptic curves. This work is contained in his thesis (https://http://eprintsphd.biblio.unitn.it/3769/1/PhD_Thesis_Rapa.pdf)

The notion of pure-injective (also known under the name "algebraically compact") module originated as a key concept in model theory of modules and extends to categories sheaves. Coherent sheaves on smooth projective schemes over a field are pure-injective, as are finite dimensional modules over finite-dimensional algebras, but the conversei s usually not true. In some sense, pure-injective quasi coherent sheaves should be the next best understandable sheaves to coherent ones, analogously to the case of modules.

The situation is completely understood for the projective line, and here we focus on elliptic curves. Building on the work of Reiten and Ringel (who introduced the notion of slope for any indecomposable quasi-coherent sheaf) and in the context of a recent work of Kussin and Laking (arXiv:1911.02485), we found large families of indecomposable pure-injective quasi-coherent sheaves of irrational slope.

Meeting-Link:

**22.06.2020**

**Thomas Eckl (Liverpool) **(Webex talk--details below)

**Nagata's Conjecture, Kaehler packings and toric degenerations**

**Abstract:** Interpolation problems in the (complex projective) plane are a classical topic still posing many wide open conjectures, like Nagata's Conjecture. In modern terminology, these conjectures can be interpreted as predictions on local positivity on polarized algebraic varieties. In this talk, we first motivate that measuring local positivity is closely connected to solving packing problems in symplectic and Kaehler geometry. In particular, we show that Nagata's Conjecture is equivalent to a prediction on the maximal possible size of multiple balls packed into the complex projective plane respecting the Kaehler structure. Finally, we indicate how toric degenerations can be used to construct Kaehler packings explicitly.

Meeting-Link:

**29.06.2020**

**Hal Schenck (Auburn) **(Webex talk--details below)

**Milnor ring of a singular projective hypersurface V(F)**

**Abstract: **For a reduced hypersurface V(f) in P^n of degree d, the Milnor algebra M(f) is a quotient of the polynomial ring by the Jacobian ideal of f. The Castelnuovo-Mumford regularity of the Milnor algebra M(f) is well understood when V(f) is smooth, as well as when V(f) has isolated singularities. We study the regularity of M(f) when V(f) has a positive dimensional singular locus. In certain situations, we prove that the regularity is bounded by (d?2)(n+1), which is the degree of the Hessian polynomial of f. However, this is not always the case, and we prove that the regularity can grow quadratically in d. I will also describe several famous theorems about why one cares about M(f), including results from hyperplane arrangement cohomology and Hodge theory

Meeting-Link:

**13.07.2020 **

**Nathan Ilten (SFU, Burnaby, Canada)** (Webex talk--details below)

**Fano schemes for complete intersections in toric varieties**

**Abstract:** The study of the set of lines contained in a fixed hypersurface is classical:Cayley and Salmon showed in 1849 that a smooth cubic surface contains 27 lines, and Schubert showed in 1879 that a generic quantic threefold contains 2875 lines. More generally, the set of k-dimensional linear spaces contained in a fixed projective variety X itself is called the k-th Fano scheme of X. These Fano schemes have been studied extensively when X is a general hypersurface or complete intersection in projective space. In this talk, I will report on work with Tyler Kelly in which we study Fano schemes for hypersurfaces and complete intersections in projective toric varieties. In particular, I'll give criteria for the Fano schemes of generic complete intersections in a projective toric variety to be non-empty and of "expected dimension". Combined with some intersection theory, this can be used for enumerative problems, for example, to show that a general degree (3,3) hypersurface in the Segre embedding of P2?P2P2?P2 contains exactly 378 lines.

Meeting link::

https://sfu.zoom.us/j/95686986289?pwd=WUVKekRQV0duU2Y1V0NwSTFYQ0xPUT09

Meeting ID: 956 8698 6289

Password: 110294

**Winter Semester 2019/2020**

**Unless otherwise specified talks take place in room 119 (Arnimallee 3) at 16:15**

**Schedule**

**17.10.2019 **

**Luis Sola Conde (Trento)**

**Characterizing rational homogeneous varieties.Abstract: I**n this talk we will discuss possible characterizations of homogeneity within the framework of Fano varieties. More concretely, we will first examine the relation among the homogeneity of a Fano variety, and the positivity of its tangent bundle. In the second part of the talk, we will consider fiber bundles on rational homogeneous varieties, and study how uniformity could imply homogeneity in this setting.

24.10.2019

**Andrzej Weber (Warschau)**

**Elliptic characteristic classes of Schubert varieties and Hecke-type algebra**

We consider the homogeneous spaces G/P, where G is a complex reductive group, and P its a parabolic subgroup. We study characteristic classes of Schubert varieties in cohomology or K theory. The elliptic classes defined by Borisov and Libgober are of special interest. To compute them we need to understand the canonical divisors of Schubert varieties. This can be obtained by analysis of Bott-Samelson resolutions. As the output we obtain inductive formulas extending the classical results of Demazure, Beilinson-Gelfan-Gelfand, Lusztig and Lascoux-Schutzenberger.

This is a joint work with Shrawan Kumar and Richard Rimanyi.

**31.10.2019 **

**Eleonora Romano (Warschau)**

**Torus actions on projective manifolds: Combinatorics vs Birational Geometry**

**Abstract:** In this talk we focus on one-dimensional torus actions on complex smooth projective varieties of arbitrary dimension. On one hand, we introduce some combinatorics tools which we need in our situation, on the other hand we discuss how Birational Geometry takes his role. In particular, we review the classical adjunction theory from a new perspective given by the torus acting. Moreover we see how this new approach allows to get some classification results for special varieties called of "small bandwidth". As an application of our results, we will conclude by putting them in the context of the classification of contact Fano varieties of high dimension. This talk is a joint work with J. Wisniewski, and includes a work in progress with G. Occhetta, L. Sola' Conde and J. Wisniewski.

**14.11.2019**

**Simon Telen (KU Leuven, Belgium/TU-Berlin)**

**Numerical Root Finding via Cox Rings**

**Abstract:** In this talk, we consider the problem of solving a system of (sparse) Laurent polynomial equations defining finitely many nonsingular points in a compact toric variety. The Cox ring of this toric variety is a generalization of the homogeneous coordinate ring of projective space. We work with multiplication maps in graded pieces of this ring to generalize the eigenvalue, eigenvector theorem for root finding in affine space. We present a numerical linear algebra algorithm for computing the corresponding matrices, and from these matrices a set of homogeneous coordinates of the roots of the system. Several numerical experiments show the effectiveness of the resulting method, especially for solving (nearly) degenerate, high degree systems in small numbers of variables.

**21.11.2019**

**Ilya Smirnov (Stockholm)**

**Lech's inequality and its combinatorics. **

**Abstract:** Multiplicity and colength are two most basic invariants of an m-primary ideal of a local ring. These two invariants can differ drastically, but in 1960 Lech found that the ratio multiplicity(I)/colength(I) is bounded uniformly by the multiplicity of the ring.

If dimension is not two, this inequality is never sharp. In a joint work with Craig Huneke and Javid Validashti, we conjectured an improvement and made a certain progress.

In my talk, I will discuss these results and will give all necessary background details. I will focus on an existing connection with enumerative combinatorics and will give a combinatorial proof of Lech's inequality and explain a combinatorial form of our conjecture.

**Wednesday!!! 04.12.2019**

**Olivier Benoist (Paris)**

**Rational curves in real rationally connected varieties.**

Abstract: Can one approximate a loop drawn in the real locus of a real rationally connected variety by the real locus of a rational curve lying on it? We will give a positive answer in particular cases, including cubic hypersurfaces, intersections of two quadrics, and compactifications of homogeneous spaces. This is joint work with Olivier Wittenberg.

**23.01.2020**

**Alessio Corti (London)**

**Smoothing Gorenstein toric affine 3-folds: some Conjectures**

**Abstract.** I present a conjectural description of the smoothing components of the deformation space. I discuss context, evidence, and relation to mirror symmetry. Work with Matej Filip and Andrea Petracci.

**30.01.2020**

**No talk: NoGaGS in Hannover**

**06.02.2020**

**Irem Portakal (Magdeburg)**

**Combinatorial aspects of Kazhdan-Lusztig varieties**

**Abstract:** In this talk, we present a combinatorial introduction to Kazhdan-Lusztig (KL) varieties in terms of Rothe diagrams. This enables us to understand the so-called usual torus action on them in terms of simple directed graphs. Moreover there is a one-to-one correspondence between KL varieties and the affine neighborhoods of torus fixed points in the Schubert variety in the full flag variety. We utilize existing results about the usual torus action on the Schubert variety in order to determine the complexity of the torus action on KL varieties. This is joint work with Donten-Bury and Escobar.

**13.02.2020**

**Roser Homs Pons (Leipzig)**

**Computing minimal Gorenstein covers**

**Abstract:** We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin k-algebra A =k[[x_1,...,x_n]]/I, compute an Artin Gorenstein k-algebra G = > k[[x_1,...,x_n]]/J such that l(G)−l(A) is minimal. We approach the problem by using Macaulay’s inverse systems and a modification of the integration method for inverse systems to compute Gorenstein covers. We propose new characterizations of the minimal Gorenstein cover and present a new algorithm for the effective computation of the variety of all minimal Gorenstein covers of A for low Gorenstein colength.

**Summer Semester 2019**

**Unless otherwise specified talks take place in room 119 (Arnimallee 3) at 16:15**

**Schedule**

**!! Tuesday, 23.04.2019 at 12: 15 in room 210 (Arnimallee 3) !!**

**Duco van Straten (Mainz)**

**Continuous algebraic geometry: exploring fractional dimension.**

**Abstract:** The concept of interpolation to a continuous variable is very old, and has turned out to be very useful, the Gamma-function being an example in case. In this talk I will report on joint work with V. Golyshev on projective spaces and Grassmanians of fractional dimension and the algebraic geometry related to it.

**02.05.2019**

**Jarek Wisniewski (Warsaw)**

**Combinatorics of C* action and varieties of small bandwidth.**

**Abstract:** Motivated by LeBrun-Salamon conjecture about quaternion-Kahler manifolds we study polarized pairs (X,L) with an action of C* on the complex manifold X such that its general orbit has small degree with respect to the ample line bundle L. For this purpose we describe adjunction morphism for (X,L) using combinatorics arising from the action of C*. In passing we get a reinterpretation of classical objects in projective and birational geometry like Severi varieties and special Cremona tranformations. I will report on a joint work with Romano and Ochetta, Sola Conde.

**16.05.2019 **

**Luca Battis tella (MPI, Bonn)**

**Reduced Gromov-Witten invariants and singularities of genus one**

**Abstract:** Classical enumerative geometry produced beautiful theorems such as: every smooth cubic surface contains exactly 27 lines. The subject has been reinvigorated since the introduction of the moduli space of stable maps, which, for example, allowed Kontsevich to solve a long-standing problem: the number of rational plane curves of degree d passing through 3d-1 general points. More generally, it made it possible to answer many questions on rational curves in projective complete intersections.

Dealing with curves of higher genus is more difficult, because the moduli spaces are not so well behaved. I will explain this with an example in genus one. We will then see two different approaches to the problem: a desingularisation of the moduli space, which led to the definition of reduced invariants, due to Li-Vakil-Zinger, and alternative compactifications obtained by Viscardi by allowing the source curve to acquire a singularity of genus one. For the quintic threefold, the two approaches are put in relation in joint work with Carocci and Manolache.

If time permits, I will discuss how log geometry - a far-fetching generalisation of toric geometry which is often useful to single out and improve the main component of moduli spaces - enters the picture, both by producing natural contractions to singular curves, and allowing us to study the relative problem, i.e. a count of curves tangent to a hyperplane section. This is joint work with Nabijou and Ranganathan, based on a ground-breaking paper of Ranganathan, Santos-Parker, and Wise.

**23.05.2019**

**Matej Filip (Mainz)**

**The versal deformation for toric varieties in special lattice degrees.**

**Abstract: **I will describe the versal deformation for non-solated Gorenstein toric varieties in the Gorenstein lattice degree.

**06.06.2019**

**Klaus Altmann (FU-Berlin)**

**Deformation of toric singularities by universal extensions of semigroups**

**Abstract:** We show how extensions of semigroups lead to deformations of toric singularities. On the level of semigroups, universal extensions exist, and they lead to versal deformations of the toric singularities in a prescribed multidegree. This is joint work with Alexandru Constantinescu (FUB) and Matej Filip (Mainz).

**20.06.2019**

**Christian Sevenheck (Chemnitz)**

**Hodge ideals for certain free divisors**

**Abstract:** In recent years, there has been renewed interest in the theory of mixed Hodge modules, mainly motivated by questions from birational geometry. In particular, Mustata and Popa have defined the so-called Hodge ideals, which describes to a certain extend to Hodge filtration on the complement of a singular divisor. In this talk, I will explain what Hodge ideals are and then discuss a particular case, namely that of certain free divisors, where quite explicit statements about the Hodge filtration can be made. This is joint work with Luis Narvaez Macarro (Sevilla) and Alberto Castaño Domínguez (Santiago de Compostella).

**Rekha Thomas (Seattle)**

**Toric slack ideals and psd-minimal polytopes**

**Abstract:** Slack ideals of polytopes are saturated determinantal ideals that encode all realizations of polytopes in the same combinatorial class. The simplest of these ideals are toric and it is an open question as to which polytopes give rise to toric slack ideals. However, we do know many special properties of polytopes with toric slack ideals and in this talk I will explain these results. They are related to both projectively unique polytopes and polytopes that admit the smallest possible lifts into psd cones, two rather unrelated concepts it seems. There are several open questions in this circle of ideas which I will mention.