Fitness, Apprenticeship, and Polynomials

arXiv:1612.03539v1 [math.AG] 12 Dec 2016

Bernd Sturmfels

Abstract This article discusses the design of the Apprenticeship Program at the
Fields Institute, held 21 August–3 September 2016. Six themes from combinatorial
algebraic geometry were selected for the two weeks: curves, surfaces, Grassmannians, convexity, abelian combinatorics, parameters and moduli. The activities were
structured into fitness, research and scholarship. Combinatorics and concrete computations with polynomials (and theta functions) empowers young scholars in algebraic geometry, and it helps them to connect with the historic roots of their field. We
illustrate our perspective for the threefold obtained by blowing up six points in P3 .

1 Design
A thematic program on Combinatorial Algebraic Geometry took place at the Fields
Institute, Toronto, Canada, during the Fall Semester 2016. The program organizers
were David Cox, Megumi Harada, Diane Maclagan, Gregory Smith, and Ravi Vakil.
As part of this semester, the Clay Mathematics Institute funded the “Apprenticeship Weeks”, held 21 August–3 September 2016. This article discusses the design
and mathematical scope of this fortnight. The structured activities took place in the
mornings and afternoons on Monday, Wednesday, and Friday, as well as the mornings on Tuesday and Thursday. The posted schedule was identical for both weeks:
MWF 9:00–9:30: Introduction to today’s theme
MWF 9:30–11:15: Working on fitness problems
MWF 11:15–12:15: Solutions to fitness problems
MWF 14:00–14:30: Dividing into research teams
MWF 14:30–17:00: Team work on projects
Bernd Sturmfels
Department of Mathematics, University of California, Berkeley, CA, 94720, United States of
America and Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103
Leipzig, Germany, e-mail: bernd@berkeley.edu or bernd@mis.mpg.edu

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Bernd Sturmfels

MWF 17:00–18:00: Teams present findings
TuTh 9:00–12:00: Discussion of the scholarship theme
The term “fitness” is an allusion to physical exercise. In order to improve physical
fitness, many of us go to the gym. A personal trainer can greatly enhance that experience. The trainer develops your exercise plan and he pushes you beyond previously
perceived limits. The trainer makes you sweat a lot, he ensures that you use exercise
equipment correctly, and he helps you to feel good about yourself afterwards. In the
context of team sports, the coach plays that role. She works towards the fitness of
the entire team, where every player will contribute to the best of their abilities.
The six fitness sessions were designed to be as intense as those in sports. Ten
problems were posted for each session, and these were available online two or three
days in advance. By design, these demanding problems were open-ended and probed
a different aspect of the theme. Section 3 of this article contains the complete list of
problems, along with a brief discussion and references that contain some solutions.
The “apprentices” were about 40 early-career mathematicians, graduate students
and postdocs, coming from a wide range of backgrounds. An essential feature of the
Apprenticeship Weeks was the effort to build teams, and to promote collaboration
as much as possible. This created an amazing sense of community within the group.
At 9:00am on each Monday, Wednesday or Friday, a brief introduction was given
to each fitness question. We formed ten teams to work on the problems. At 11:15am
we got together again, and one person from each team gave a brief presentation on
what had been discussed and discovered. Working on a challenging problem, with
a group of new collaborators, for less than two hours created a very intense and
stimulating experience. A balanced selection process ensured that each participant
had the opportunity to present for their team at least once.
At 2:00pm the entire group re-assembled and they discussed research-oriented
problems for the afternoons. This was conducted in the style of the American Institute for Mathematics (AIM), whereby one of the participants serves as the discussion leader, and only that person is allowed to touch the blackboard. This led to
an ample supply of excellent questions, some a direct continuation of the morning
fitness problems, and others only vaguely inspired by these. Again, groups were
formed for the afternoon, and they engaged in learning and research. Computations
and literature search played a big role, and a lot of teaching went on in the groups.
Tuesday and Thursdays were discussion days. Here the aim was to create a sense
of scholarship among the participants. The morning of these days involved studying
various software packages, classical research papers from the 19-th and early 20th centuries, and the diverse applications of combinatorial algebraic geometry. The
prompts are given in Section 2. The afternoons on discussion days were unstructured
to allow the participants time to ponder, probe, and write up their many new ideas.

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2 Scholarship Prompts
Combinatorial algebraic geometry is a field that, by design, straddles mathematical boundaries. One aim is to study algebraic varieties with special combinatorial
features. At its roots, this field is about systems of polynomial equations in several
variables, and about symmetries and other special structures in their solution sets.
Section 5 offers a concrete illustration of this perspective for a system of polynomials in 32 variables. The objects of combinatorial algebraic geometry are amenable
to a wide range of software tools, which are now used widely among the researchers.
Another point we discussed is the connection to problems outside of pure mathematics. A new field, Applied Algebraic Geometry, has arisen in the past decade. The
techniques used there often connect back to 19th and early 20th century work in algebraic geometry, which is much more concrete and combinatorial than many recent
developments. And, even for her study of current abstract theories, an apprentice
may benefit from knowing the historic origins that have inspired the development of
algebraic geometry. Understanding these aspects, by getting hands-on experiences
and by studying original sources, was a focus in this part of the program.
In what follows we replicate the hand-outs for the four TuTh mornings. The
common thread can be summarized as: back to the roots. These were given to the
participants as prompts for explorations and discussions. For several of the participants, it was their first experience with software for algebraic geometry. For others,
it offered a first opportunity to read an article that was published over 100 years ago.

Tuesday, August 23: Software
Which software tools are most useful for performing computations in
Combinatorial Algebraic Geometry ? Why?
Many of us are familiar with Macaulay2. Some of us are familiar with
Singular. What are your favorite packages within these systems?
Lots of math is supported by general-purpose computer algebra systems such as
Sage, Maple, Mathematica, or Magma. Do you use any of these regularly? For
research or for teaching? How often and in which context?
Other packages that are useful for our community include Bertini, PHCpack,
4ti2, Polymake, Normaliz, GFan. What are these and what do they do? Who
developed them and why?
Does visualization matter in algebraic geometry?
Have you tried software like Surfex?
Which software tool do you want to learn today?

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Thursday, August 25: The 19th Century
Algebraic Geometry has a deep and distinguished history that goes back hundreds
of years. Combinatorics entered the scene a bit more recently.
Young scholars interested in algebraic geometry are strongly encouraged to familiarize themselves with the literature from the 19th century. Dig out papers from
that period and read them! Go for the original sources. Some are in English. Do not
be afraid of languages like French, German, Italian.
Today we form groups. Each group will explore the life and work of one mathematician, with focus on what he has done in algebraic geometry. Identify one key
paper written by that author. Then present your findings.
Here are some suggestions, listed alphabetically:
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Alexander von Brill
Arthur Cayley
Michel Chasles
Luigi Cremona
Georges Halphen
Otto Hesse
Ernst Kummer
Max Noether
Julius Plücker
Bernhard Riemann
Friedrich Schottky
Hermann Schubert
Hieronymus Zeuthen

Tuesday, August 30: Applications
The recent years have seen a lot of interest in applications of algebraic geometry,
outside of core pure mathematics. An influential event was a research year 200607 at the IMA in Minneapolis. Following a suggestion by Doug Arnold (then IMA
director and SIAM president), it led to the creation of the SIAM activity group
in Algebraic Geometry, and (ultimately) to the SIAM Journal on Applied Algebra
and Geometry. The reader is referred to these resources for further information.
These interactions with the sciences and engineering have been greatly enhanced by
the interplay with Combinatorics and Computation seen here at the Fields Institute.
However, the term “Algebraic Geometry” has to be understood now in a broad sense.
Today we form groups. Each group will get familiar with one field of application,
and they will select one paper in Applied Algebraic Geometry that represent an
interaction with that field. Read your paper and then present your findings. Here are
some suggested fields, listed alphabetically:
• Approximation Theory

Fitness, Apprenticeship, and Polynomials

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Bayesian Statistics
Chemical Reaction Networks
Coding Theory
Combinatorial Optimization
Computer Vision
Cryptography
Game Theory
Geometric Modeling
Machine Learning
Maximum Likelihood Inference
Neuroscience
Phylogenetics
Quantum Computing
Semidefinite Programming
Systems Biology

Thursday, September 1: The Early 20th Century
One week ago we examined the work of some algebraic geometers from the 19th
century. Today, we move on to the early 20th century, to mathematics that was published prior to World War II. You are encouraged to familiarize yourselves with
the literature from the period 1900-1939. Dig out papers from that period and read
them! Go for the original sources. Some are written in English. Do not be afraid of
languages like French, German, Italian, Russian.
Each group will explore the life and work of one mathematician, with focus on
what (s)he has done in algebraic geometry during that period. Identify one key paper
written by that author. Then present your findings.
Here are some suggestions, listed alphabetically:
•
•
•
•
•
•
•
•
•
•
•
•
•
•
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Eugenio Bertini
Guido Castelnuovo
Wei-Liang Chow
Arthur B. Coble
Wolfgang Gröbner
William V.D. Hodge
Wolfgang Krull
Solomon Lefschetz
Frank Morley
Francis S. Macaulay
Amalie Emmy Noether
Ivan Georgievich Petrovsky
Virginia Ragsdale
Gaetano Scorza
Francesco Severi

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3 Fitness Prompts
This section presents the six worksheets for the morning sessions on Mondays,
Wednesdays and Fridays. These prompts inspired most of the articles in this volume. Specific pointers to dates refer to events that took place at the Fields Institute.
The next section contains notes for each problem, offering references and solutions.

Monday, August 22: Curves
1. Which genus can a smooth curve of degree 6 in P3 have? Give examples.
2. Let f (x) = (x − 1)(x − 2)(x − 3)(x − 6)(x − 7)(x − 8) and consider the genus
2 curve y2 = f (x). Where is it in the moduli space M2 ? Compute the Igusa
invariants. Draw the Berkovich skeleton for the field of 5-adic numbers.
3. The tact invariant of two plane conics is the polynomial of bidegree (6, 6) in
the 6 + 6 coefficients which vanishes when the conics are tangent. Compute this
invariant explicitly. How many terms does it have?
4. Bring’s curve lives in a hyperplane in P4 . It is defined by xi0 +xi1 +xi2 +xi3 +xi4 = 0
for i = 1, 2, 3. What is its genus? Determine all tritangent planes of this curve.
5. Let X be a curve of degree d and genus g in P3 . The Chow form of X defines a
hypersurface in the Grassmannian Gr(1, P3 ). Points are lines that meet X. Find
the dimension and (bi)degree of its singular locus.
6. What are the equations of the secant varieties of elliptic normal curves?
7. Let XP be the toric variety defined by a 3-dimensional lattice polytope, as in
Milena Hering’s July 18-22 course. Intersect XP with two general hyperplanes to
get a curve. What is the degree and genus of that curve?
8. A 2009 article by Sean Keel and Jenia Tevelev presents Equations for M 0,n .
Write these equations in Macaulay2 format for n = 5 and n = 6. Can you see
the ψ -classes (seen in Renzo Cavalieri’s July 18-22 course) in these coordinates?
9. Review the statement of Torelli’s Theorem for genus 3. Using Sage or Maple,
compute the 3 × 3 Riemann matrix of the Fermat quartic {x4 + y4 + z4 = 0}. How
can you recover the curve from that matrix?
10. The moduli space M7 of genus 7 curves has dimension 18. What is the codimension of the locus of plane curves? Hint: Singularities are allowed.

Wednesday, August 24: Surfaces
1. A nondegenerate surface in Pn has degree at least n − 1. Prove this fact and determine all surfaces of degree n − 1. Give their equations.
2. How many lines lie on a surface obtained by intersecting two quadratic hypersurfaces in P4 ? Find an instance where all lines are defined over Q.
3. What is the maximum number of singular points on an irreducible quartic surface
in P3 ? Find a surface and compute its projective dual.

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4. Given a general surface of degree d in P3 , the set of its bitangent lines is a surface
in Gr(1, P3 ). Determine the cohomology class (or bidegree) of that surface.
5. Pick two random circles C1 and C2 in R3 . Compute their Minkowski sum C1 +C2
and their Hadamard product C1 ⋆ C2 . Try other curves.
6. Let X be the surface obtained by blowing up five general points in the plane.
Compute the Cox ring of X. Which of its ideals describe points on X?
7. The incidences among the 27 lines on a cubic surface defines a 10-regular graph.
Compute the complex of independent sets in this graph.
8. The Hilbert scheme of points on a smooth surface is smooth. Why? How many
torus-fixed points are there on the Hilbert scheme of 20 points in P2 ? What can
you say about the graph that connects them?
9. State the Hodge Index Theorem. Verify this theorem for cubic surfaces in P3 , by
explicitly computing the matrix for the intersection pairing.
10. List the equations of one Enriques surface. Verify its Hodge diamond.

Friday, August 26: Grassmannians
1. Find a point in Gr(3, 6) with precisely 16 non-zero Plücker coordinates. As in
June Huh’s July 18-22 course, determine the Chow ring of its matroid.
2. The coordinate ring of the Grassmannian Gr(3, 6) is a cluster algebra of finite
type. What are the cluster variables? List all the clusters.
3. Consider two general surfaces in P3 whose degrees are d and e respectively. How
many lines in P3 are bitangent to both surfaces?
4. The rotation group SO(n) is an affine variety in the space of real n × n-matrices.
Can you find a formula for the degree of this variety?
5. The complete flag variety for GL(4) is a six-dimensional subvariety of P3 × P5 ×
P3 . Compute its ideal and determine its tropicalization.
6. Classify all toric ideals that arises as initial ideals for the flag variety above. For
each such toric degeneration, compute the Newton-Okounkov body.
7. The Grassmannian Gr(4, 7) has dimension 12. Four Schubert cycles of codimension 3 intersect in a finite number of points. How large can that number be?
Exhibit explicit cycles whose intersection is reduced.
8. The affine Grassmannian and the Sato Grassmannian are two infinite-dimensional
versions of the Grassmannian. How are they related?
9. The coordinate ring of the Grassmannian Gr(2, 7) is Z7 -graded. Determine the
Hilbert series and the multidegree of Gr(2, 7) for this grading.
10. The Lagrangian Grassmannian parametrizes n-dimensional isotropic subspaces
in C2n . Find a Gröbner basis for its ideal. What is a ‘doset’?

Monday, August 29: Convexity
1. The set of nonnegative binary sextics is a closed full-dimensional convex cone in
Sym6 (R2 ) ≃ R7 . Determine the face poset of this convex cone.

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2. Consider smooth projective toric fourfolds with eight invariant divisors. What is
the maximal number of torus-fixed points of any such variety?
3. Choose three general ellipsoids in R3 and compute the convex hull of their union.
Which algebraic surfaces contribute to the boundary?
4. Explain how the Alexandrov-Fenchel Inequalities (for convex bodies) can be derived from the Hodge Index Theorem (for algebraic surfaces).
5. The blow-up of P3 at six general points is a threefold that contains 32 special
surfaces (exceptional classes). What are these surfaces? Which triples intersect?
Hint: Find a 6-dimensional polytope that describes the combinatorics.
6. Prove that every face of a spectrahedron is an exposed face.
7. How many combinatorial types of reflexive polytopes are there in dimension 3?
In dimension 4? Draw pictures of some extreme specimen.
8. A 4 × 4-matrix has six off-diagonal 2 × 2-minors. Their binomial ideal in 12
variables has a unique toric component. Determine the f-vector of the polytope
(with 12 vertices) associated with this toric variety.
9. Consider the Plücker embedding of the real Grassmannian Gr(2, 5) in the unit
sphere in R10 . Describe its convex hull. Hint: Calibrations, Orbitopes.
10. Examine Minkowski sums of three tetrahedra in R3 . What is the maximum number of vertices such a polytope can have? How to generalize?

Wednesday, August 31: Abelian Combinatorics
1. The intersection of two quadratic surfaces in P3 is an elliptic curve. Explain its
group structure in terms of geometric operations in P3 .
2. A 2006 paper by Keiichi Gunji gives explicit equations for all abelian surfaces
in P8 . Verify his equations in Macaulay2. How to find the group law?
3. Experiment with Swierczewski’s Sage code for the numerical evaluation of the
Riemann theta function θ (τ ; z). Verify the functional equation.
4. Theta functions with characteristics θ [ε , ε ′ ](τ ; z) are indexed by two binary vectors ε , ε ′ ∈ {0, 1}g. They are odd or even. How many each?
5. Fix the symplectic form hx, yi = x1 y4 + x2 y5 + x3 y6 + x4 y1 + x5 y2 + x6 y3 on the
64-element vector space (F2 )6 . Determine all isotropic subspaces.
6. Explain the combinatorics of the root system of type E7 . How would you choose
coordinates? How many pairs of roots are orthogonal?
7. In 1879 Cayley published a paper in Crelle’s journal titled Algorithms for ...
What did he do? How does it relate the previous two exercises?
8. The regular matroid R10 defines a degeneration of abelian 5-folds. Describe its
periodic tiling on R5 and secondary cone in the 2-nd Voronoi decomposition.
Explain the application to Prym varieties due to Gwena.
9. Consider the Jacobian of the plane quartic curve defined over Q2 by

Fitness, Apprenticeship, and Polynomials

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41x4 + 1530x3y + 3508x3z + 1424x2y2 + 2490x2yz
− 2274x2z2 + 470xy3 + 680xy2z − 930xyz2 + 772xz3
+ 535y4 − 350y3z − 1960y2z2 − 3090yz3 − 2047z4
Compute its limit in Alexeev’s moduli space for the 2-adic valuation.
10. Let Θ be the theta divisor on an abelian threefold X. Find n = dim H 0 (X, kΘ ).
What is the smallest integer k such that kΘ is very ample? Can you compute (in
Macaulay2) the ideal of the corresponding embedding X ֒→ Pn−1 ?

Friday, September 2: Parameters and Moduli
1. Write down (in Macaulay2 format) the two generators of the ring of invariants
for ternary cubics. For which plane cubics do both invariants vanish?
2. Fix a Z-grading on the polynomial ring S = C[a, b, c, d] defined by deg(a) = 1,
deg(b) = 4, deg(c) = 5, and deg(d) = 9. Classify all homogeneous ideals I such
that S/I has Hilbert function identically equal to 1.
3. Consider the Hilbert scheme of eight points in affine 4-space A4 . Identify a point
that is not in the main component. List its ideal generators.
4. Let X be the set of all symmetric 4 × 4-matrices in R4×4 that have an eigenvalue
of multiplicity ≥ 2. Compute the C-Zariski closure of X.
5. Which cubic surfaces in P3 are stable? Which ones are semi-stable?
6. In his second lecture on August 15, Valery Alexeev used six lines in P2 to construct a certain moduli space of K3 surfaces with 15 singular points. List the most
degenerate points in the boundary of that space.
7. Find the most singular point on the Hilbert scheme of 16 points in A3 .
8. The polynomial ring C[x, y] is graded by the 2-element group Z/2Z where
deg(x) = 1 and deg(y) = 1. Classify all Hilbert functions of homogeneous ideals.
9. Consider all threefolds obtained by blowing up six general points in P3 . Describe
their Cox rings and Cox ideals. How can you compactify this moduli space?
10. The moduli space of tropical curves of genus 5 is a polyhedral space of dimension
12. Determine the number of i-faces for i = 0, 1, 2, . . . , 12.

4 Notes, Solutions and References
Solutions to several of the sixty fitness problems can be found in the 16 articles of
this volume. The articles are listed as the first 16 entries in our References. They
will be published in the order in which they are cited in this section. In what follows
we also offer references for other problems that did not lead to articles in this book.

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Notes on Curves
1. Castelnuovo classified the degree and genus pairs (d, g) for all smooth curves in
Pn . This was extended to characteristic p by Ciliberto [25]. For n = 3, d = 6, the
possible genera are g = 0, 1, 2, 3, 4. The Macaulay2 package RandomCurves
can compute examples. The Hartshorne-Rao module [50] plays a key role.
2. See Section 2 in the article by Bolognese, Brandt and Chua [1]. The approach
using Igusa invariants was developed by Helminck in [32].
3. The tact invariant has 3210 terms, by [57, Example 2.7].
4. See Section 2.1 in the article by Harris and Len [2]. The analogous problem for
bitangents of plane quartics is discussed by Chan and Jiradilok [3].
5. This is solved in the article by Kohn, Nødland and Tripoli [4]
6. Following Fisher [29], elliptic normal curves are defined by the 4 × 4-subpfaffians
of the Klein matrix, and their secant varieties are defined by its larger subpfaffians.
7. The degree of a projective toric variety XP is the volume of its lattice polytope P.
The genus of a complete intersection in XP was derived by Khovanskii in 1978. We
recommend the tropical perspective offered by Steffens and Theobald in [53, §4.1].
8. See the article by Monin and Rana [5] for a solution up to n = 6.
9. See [26] for how to compute the forward direction of the Torelli map of an arbitrary plane curve. For computing the backward direction in genus 3 see [61, §5.2].
10. Trinodal sextics form a 16-dimensional family; their codimension in M7 is two.
This is a result due to Severi, derived by Castryck and Voight in [24, Theorem 2.1].

Notes on Surfaces
1. This was solved by Del Pezzo in 1886. Eisenbud and Harris [27] give a beautiful
introduction to the theory of varieties of minimal degree, including their equations.
2. This is a del Pezzo surface of degree 4. It has 16 lines. To make them rational,
map P2 into P3 via a Q-basis for the cubics that vanish at five rational points in P2 .
3. The winner, with 16 singular points, is the Kummer surface [34]. It is self-dual.
4. This is solved in the article by Kohn, Nødland and Tripoli [4].
5. See Section 5 in the article by Friedenberg, Oneto and Williams [6].
6. This is the del Pezzo surface in Problem 2. Its Cox ring is a polynomial ring in
16 variables modulo an ideal generated by 20 quadrics. Ideal generators that are
universal over the base M0,5 are listed in [47, Proposition 2.1]. Ideals of points on
the surface are torus translates of the toric ideal of the 5-dimensional demicube D5 .
For six points in P2 we refer to Bernal, Corey, Donten-Bury, Fujita and Merz [7].
7. This is the clique complex of the Schläfli graph. The f-vector of this simplicial
complex is (27, 216, 720, 1080, 648, 72). The Schläfli graph is the edge graph of the
E6 -polytope, denoted 221 , which is a cross section of the Mori cone of the surface.
8. The torus-fixed points on Hilb20 (P2 ) are indexed by ordered triples of partitions

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(λ1 , λ2 , λ3 ) with |λ1 | + |λ2 | + |λ3 | = 20. The number of such triples equals 341, 649.
The graph that connects them is a variant of the graph for the Hilbert scheme of
points in the affine plane. The latter was studied by Hering and Maclagan in [33].
9. The signature of the intersection pairing is (1, r − 1) where r is the rank of the
Picard group. This is r = 7 for the cubic surface. From the analysis in Problem 7,
we can get various symmetric matrices that represent the intersection pairing.
10. See the article by Bolognese, Harris and Jelisiejew [8].

Notes on Grassmannians
1. See the article by Wiltshire-Gordon, Woo and Zajackowska [9].
2. In addition to the 20 Plücker coordinates pi jk , one needs two more functions,
namely p123 p456 − p124 p356 and p234 p561 − p235 p461 . The six boundary Plücker coordinates p123 , p234 , p345 , p456 , p561 , p612 are frozen. The other 16 coordinates are
the cluster variables for Gr(3, 6). This was derived by Scott in [51, Theorem 6].
3. This is worked out in the article by Kohn, Nødland and Tripoli [4].
4. This is the main result of Brandt, Bruce, Brysiewicz, Krone and Robeva [10].
5. See the article by Bossinger, Lamboglia, Mincheva and Mohammadi [11].
6. See the article by Bossinger, Lamboglia, Mincheva, Mohammadi [11].
7. The maximum number is 8. This is obtained by taking the partition (2, 1) four
times. For this problem, and many other Schubert problems, instances exist where
all solutions are real. See the works of Sottile, specifically [52, Theorem 3.9 (iv)].
8. The Sato Grassmannian is more general than the affine Grassmannian. These are
studied, respectively, in integrable systems and in geometric representation theory.
9. A formula for the Zn -graded Hilbert series of Gr(2, n) is given by Witaszek [63,
§3.3]. For an introduction to multidegrees see [40, §8.5]. Try the Macaulay2 commands Grassmannian and multidegree. Escobar and Knutson [12] determine the multidegree of a variety that is important in computer vision.
10. The coordinate ring of the Lagrangian Grassmannian is an algebra with straightening law over a doset. This stands for double poset. See the exposition in [48, §3].

Notes on Convexity
1. The face lattice of the cone of non-negative binary forms of degree d is described
in Barvinok’s textbook [20, §II.11]. In more variables this is much more difficult.
2. This seems to be an open problem. For seven invariant divisors, this was resolved
by Gretenkort et al. [30]. Note the conjecture stated in the last line of that paper.
3. We refer to Nash, Pir, Sottile and Ying [13] and to the youtube video The Convex Hull of Ellipsoids by Nicola Geismann, Michael Hemmer, and Elmar Schömer.
4. We refer to Ewald’s textbook, specifically [28, §IV.5 and §VII.6].

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5. The relevant polytope is the 6-dimensional demicube; its 32 vertices correspond
to the 32 special divisors. See the notes for Problem 9 in Parameters and Moduli.
6. This was first proved by Ramana and Goldman in [43, Corollary 1].
7. Kreuzer and Skarke [37] classified such reflexive polytopes up to lattice isomorphism. There are 4319 in dimension 3, and there are 473800776 in dimension 4.
Lars Kastner classified the list of 4319 into combinatorial types. He found that there
are 558 combinatorial types of reflexive 3-polytopes. They have up to 14 vertices.
8. This 6-dimensional polytope is obtained from the direct product of two identical regular tetrahedra by removing the four pairs of corresponding vertices. It is the
convex hull of the points ei ⊕ e j in R4 ⊕ R4 where i, j ∈ {1, 2, 3, 4} with i 6= j. Using
the software Polymake, we find its f-vector to be (12, 54, 110, 108, 52, 12).
9. The faces of the Grassmann orbitopes conv(Gr(2, n)) for n ≥ 5 are described in
[49, Theorem 7.3]. It is best to start with the easier case n = 4 in [49, Example 7.1].
10. The maximum number of vertices is 38, by the formula of Karavelas et al. in
[35, §6.1, equation (49)]. A definitive solution to the problem of characterizing face
numbers of Minkowski sums of polytopes was given by Adiprasito and Sanyal [17].

Notes on Abelian Combinatorics
1. A beautiful solution was written up by Qiaochu Yuan when he was a high school
student; see [62]. The idea is to simultaneously diagonalize the two quadrics, then
project their intersection curve into the plane, thereby obtaining an Edwards curve.
2. This is a system of 9 quadrics and 3 cubics, derived from Coble’s cubic as in [45,
Theorem 3.2]. Using theta functions as in [45, Lemma 3.3], one gets the group law.
3. See [61] and compare with Problem 9 in Curves.
4. For the 22g pairs (ε , ε ′ ), we check whether ε · ε ′ is even or odd. There are
2g−1 (2g + 1) even theta characteristics and 2g−1 (2g − 1) odd theta characteristics.
5. The number of isotropic subspaces of (F2 )6 is 63 of dimension 1, it is 315 in
dimension 2, and it is 135 in dimension 3. The latter are the Lagrangians [46, §6].
6. The root system of type E7 has 63 positive roots. They are discussed in [46, §6].
7. Cayley gives a bijection between the 63 positive roots of E7 with the 63 non-zero
vectors in (F2 )6 . Two roots have inner product zero if and only if the corresponding
vectors in (F2 )6 \{0} are orthogonal in the setting of Problem 5. See [46, Table 1].
8. This refers to Gwena’s article [31]. Since the matroid R10 is not co-graphic, the
corresponding tropical abelian varieties are not in the Schottky locus of Jacobians.
9. This fitness problem is solved in the article by Bolognese, Brandt and Chua [1]
Chan and Jiradilok [3] study an important special family of plane quartics.
10. The divisor kΘ is very ample for k = 3. This embeds any abelian threefold into
P26 . For products of three cubic curves, each in P2 , this gives the Segre embedding.

Fitness, Apprenticeship, and Polynomials

13

Notes on Parameters and Moduli
1. The solution can be found, for instance, on the website
http://math.stanford.edu/∼notzeb/aronhold.html
The two generators have degree 4 and 6. The quartic invariant is known as the Aronhold invariant and it vanishes when the ternary cubic is a sum of three cubes of
linear forms. Both invariants vanish when the cubic curve has a cusp.
2. This refers to extra irreducible components in toric Hilbert schemes [42]. These
schemes were first introduced by Arnold [19], who coined the term A-graded algebras. Theorem 10.4 in [54] established the existence of an extra component for
A = (1347). We ask to verify the second entry in Table 10-1 on page 88 of [54].
3. Cartwright et al. [22] showed that the Hilbert scheme of eight points in A4 has
two irreducible components. An explicit point in the non-smoothable component is
given in the article by Douvropoulos, Jelisiejew, Nødland and Teitler [14].
4. At first, it is surprising that X has codimension 2. The point is that we work
over the real numbers R. The analogous set over C is the hypersurface of a sum-ofsquares polynomial. The C-Zariski closure of X is a nice variety of codimension 2.
The defining ideal and its Hilbert-Burch resolution are explained in [56, §7.1].
5. This is an exercise in Geometric Invariant Theory [41]. A cubic surface is stable if
and only if it has at most ordinary double points (A1 singularities). For semi-stable
surfaces, A2 singularities are allowed. For an exposition see [44, Theorem 3.6]; this
is E. Reinecke’s Bachelor thesis, written under the supervision of D. Huybrechts.
6. This is the moduli space of stable hyperplane arrangements [18], here for the case
of six lines in P2 . The precise space depends on a choice of parameters [18, §5.7].
For some natural parameters, this is the tropical compactification associated with
the tropical Grassmannian Gr(3, 6), so the most degenerate points correspond to the
seven generic types of tropical planes in 5-space, shown in [38, Figure 5.4.1].
7. See [55, Theorem 2.3].
8. For each partition, representing a monomial ideal in C[x, y], we count the odd and
even boxes in its Young diagram. The resulting Hilbert functions h : Z/2Z → N are
(h(even), h(odd)) = (k2 + m, k(k + 1) + m) or ((k + 1)2 + m, k(k + 1) + m), where
k, m ∈ N. This was contributed by Dori Bejleri. For more details see [21, §1.3].
9. The blow-up of Pn−3 at n points is a Mori dream space. Its Cox ring has 2n−1
generators, constructed explicitly by Castravet and Tevelev in [23]. These form a
Khovanskii basis [36], by [60, Theorem 7.10]. The Cox ideal is studied in [59].
Each point on its variety represents a rank two stable quasiparabolic vector bundle
on P1 with n marked points. The relevant moduli space is M0,n .
10. The moduli space of tropical curves of genus 5 serves as the first example in
the article by Lin and Ulirsch [15]. The article by Kastner, Shaw and Winz [16]
discusses state-of-the-art software tools for computing with such polyhedral spaces.

14

Bernd Sturmfels

5 Polynomials
The author of this article holds the firm belief that algebraic geometry concerns the
study of solution sets to systems of polynomial equations. Historically, geometers
explored curves and surfaces that are zero sets of polynomials. It is the insights
gained from these basic figures that have led, over the course of centuries, to the
profound depth and remarkable breadth of contemporary algebraic geometry. However, many of the current theories are now far removed from explicit varieties, and
polynomials are nowhere in sight. What we are advocating is for algebraic geometry to take an outward-looking perspective. Our readers should be aware of the
wealth of applications in the sciences and engineering, and be open to a “back to
the basics” approach in both teaching and scholarship. From this perspective, the
interaction with combinatorics can be particularly valuable. Indeed, combinatorics
is known to some as the “nanotechnology of mathematics”. It is all about explicit
objects, those that can be counted, enumerated, and dissected with laser precision.
And, these objects include some beautiful polynomials and the ideals they generate.
The following example serves as an illustration. We work in a polynomial ring
Q[p] in 32 variables, one for each subset of {1, 2, 3, 4, 5, 6} whose cardinality is odd:
p1 , p2 , . . . , p6 , p123 , p124 , p125 , . . . , p356 , p456 , p12345, p12346 , . . . , p23456.
The polynomial ring Q[p] is Z7 -graded by setting degree(pσ ) = e0 + ∑i∈σ ei , where
e0 , e1 , . . . , e6 is the standard basis of Z7 . Let X be a 5 × 6-matrix of variables, and let
I be the kernel of the ring map Q[p] → Q[X] that takes the variables pσ to the determinant of the submatrix of X with column indices σ and row indices 1, 2, . . . , |σ |.
The ideal I is prime and Z7 -graded. It has multiple geometric interpretations.
First of all, it describes the partial flag variety of points in 2-planes in hyperplanes
in P5 . This flag variety lives in P5 × P19 × P5 , thanks to the Plücker embedding. Its
projection into the factor P19 is the Grassmannian Gr(3, 6) of 2-planes in P5 . Flag
varieties are studied by Bossinger, Lamboglia, Mincheva and Mohammadi in [11].
But, let the allure of polynomials now speak for itself. Our ideal I has 66 minimal
quadratic generators. Sixty generators are unique up to scaling in their degree:
degree
(2, 0, 0, 1, 1, 1, 1)
(2, 0, 1, 0, 1, 1, 1)
···
(2, 1, 1, 1, 1, 0, 0)

ideal generator
p3 p456 − p4 p356 + p5 p346 − p6 p345
p2 p456 − p4 p256 + p5 p246 − p6 p245
··· ··· ···
p1 p234 − p2 p134 + p3 p124 − p4 p123

(2, 0, 1, 1, 1, 1, 2)
···
(2, 2, 1, 1, 1, 1, 0)

p256 p346 − p246 p356 + p236 p456
··· ··· ···
p125 p134 − p124 p135 + p123 p145

(2, 1, 1, 1, 1, 2, 2)
···
(2, 2, 2, 1, 1, 1, 1)

p156 p23456 − p256 p13456 + p356 p12456 − p456 p12356
··· ··· ···
p123 p12456 − p124 p12356 + p125 p12346 − p126 p12345

Fitness, Apprenticeship, and Polynomials

15

The other six minimal generators live in degree (2, 1, 1, 1, 1, 1, 1). These are the 4term Grassmann-Pücker relations, like p126 p345 − p125 p346 + p124 p356 − p123 p456 .
Hereis an alternate interpretation of the ideal I. It defines a variety of dimension
15 = 62 in P31 known as the spinor variety. In this guise, I encodes the algebraic
relations among the principal subpfaffians of a skew-symmetric 6 × 6-matrix. Such
subpfaffians are indexed with the subsets of {1, 2, 3, 4, 5, 6} of even cardinality. The
trick is to fix a natural bijection between even and odd subsets. This variety is similar
to the Lagrangian Grassmannian seen in fitness problem # 10 on Grassmannians.
At this point, readers who like combinatorics and computations may study I.
Can you compute the tropical variety of I? Which of its maximal cones are prime
in the sense of Kaveh and Manon [36, Theorem 1]? These determine Khovanskii
bases for Q[p]/I and hence toric degenerations of the spinor variety in P31 . Their
combinatorics is recorded in a list of Newton-Okounkov polytopes with 32 vertices.
Each of these polytopes comes with a linear projection to the 6-dimensional
demicube, which is the convex hull in R7 of the 32 points deg(pσ ). We saw this
demicube in fitness problem # 5 on Convexity, whose theme we turn to shortly.
It is the author’s opinion that Khovanskii bases deserve more attention than the
Newton-Okounkov bodies they give rise to. The former are the algebraic manifestation of a toric degeneration. These must be computed and verified. Looking at a
Khovanskii basis through the lens of convexity reveals the Newton-Okounkov body.
We now come to a third, and even more interesting, geometric interpretation of
our 66 polynomials. It has to do with Cox rings, and their Khovanskii bases, similar
to those in the article by Bernal, Corey, Donten-Bury, Fujita and Merz. We begin by
replacing the generic 5 × 6-matrix X by one that has the special form in [23, (1.2)]:

 2
u1 x1 u22 x2 u23 x3 u24 x4 u25 x5 u26 x6
 u 1 y1 u 2 y2 u 3 y3 u 4 y4 u 5 y5 u 6 y6 



X = 
 u 1 v1 x1 u 2 v2 x2 u 3 v3 x3 u 4 v4 x4 u 5 v5 x5 u 6 v6 x6  .
 v1 y1 v2 y2 v3 y3 v4 y4 v5 y5 v6 y6 
v21 x1 v22 x2 v23 x3 v24 x4 v25 x5 v26 x6
Now, the polynomial ring Q[X] gets replaced by k[x1 , x2 , . . . , x6 , y1 , y2 , . . . , y6 ] where
k is the field extension of Q generated by the entries of a 2 × 6-matrix of scalars:


u1 u2 u3 u4 u5 u6
.
(1)
U =
v1 v2 v3 v4 v5 v6
We assume that the 2 × 2-minors of U are non-zero. Let J denote the kernel of the
odd-minors map k[p] → k[X] as before. The ideal J is also Z7 -graded and it strictly
contains the ideal I. Castravet and Tevelev [23, Theorem 1.1] proved that k[p]/J is
the Cox ring of the blow-up of P3k at six points. These points are Gale dual to U.
We refer to J as the Cox ideal of that rational threefold whose Picard group Z7 furnishes the grading. The affine variety in A32
k defined by J is 10-dimensional (it is the
universal torsor). Quotienting by a 7-dimensional torus action yields our threefold.
The same story for blowing up five points in P2k is problem # 6 on Surfaces.
In [59] we construct the Cox ideal by duplicating the ideal of the spinor variety:

16

Bernd Sturmfels

J = I + u ∗ I.

(2)

Here u is a vector in (K ∗ )32 that is derived from U. The ideal u ∗ I is obtained from I
by scaling the variables fσ with the coordinates of u. In particular, the Cox ideal J is
minimally generated by 132 quadrics. Now, there are two generators in each of the
sixty Z7 -degrees in our table, and there are 12 generators in degree (2, 1, 1, 1, 1, 1, 1).
Following [60, Example 7.6], we fix the rational function field k = Q(t) and set


1 t t2 t3 t4 t5
U =
.
t5 t4 t3 t2 t 1
The ring map k[p] → k[X] now maps the variables pσ like this:
p1 →
7 x1
p123 7→ x1 y2 x3 t 6 − (x1 x2 y3 + y1 x2 x3 )t 7 + (y1 x2 x3 + x1 x2 y3 )t 9 − x1y2 x3 t 10
p12345 7→ x1 y2 x3 y4 x5 t 10 − (y1 x2 x3 y4 x5 + x1y2 x3 x4 y5 + · · · + x1 x2 y3 y4 x5 )t 11 + · · ·
Here is a typical example of a Z7 -degree with two minimal ideal generators:
(2, 1, 1, 1, 1, 0, 0)
(2, 1, 1, 1, 1, 0, 0)

p1 p234 − p2 p134 + p3 p124 − p4 p123
t4 p

1 p234 − t

2p

2 p134 + t

2p

1 p234 + p4 p123

The algebra generators pσ form a Khovanskii basis for k[p]/J with respect to the
t-adic valuation. The toric algebra resulting from this flat family is generated by the
underlined monomials. Its toric ideal in(J) is generated by 132 binomial quadrics:
degree
(2, 0, 0, 1, 1, 1, 1)
(2, 0, 1, 0, 1, 1, 1)
···
(2, 1, 1, 1, 1, 0, 0)
(2, 0, 1, 1, 1, 1, 2)
···
(2, 2, 1, 1, 1, 1, 0)
(2, 1, 1, 1, 1, 2, 2)

pair of binomial generators for in(J)
p3 p456 − p4 p356
p5 p346 − p6 p345
p2 p456 − p4 p256
p5 p246 − p6 p245
···
···
p1 p234 − p2 p134
p3 p124 − p4 p123
p6 p23456 − p236 p456
p246 p356 − p256 p346
···
···
p1 p12345 − p123 p145
p124 p135 − p125 p134
p156 p23456 − p256 p13456 p356 p12456 − p456 p12356

···
(2, 2, 2, 1, 1, 1, 1)

···
p123 p12456 − p124 p12356

···
p125 p12346 − p126 p12345

These 132 binomials define a toric variety that is a degeneration of our universal
torsor. The ideal in(J) is relevant in both biology and physics. It represents the
Jukes-Cantor model in phylogenetics [58] and the Wess-Zumino-Witten model in
conformal field theory [39]. Beautiful polynomials can bring the sciences together.
Let us turn to another fitness problem. The past three pages offered a capoeira
approach to # 9 in Parameters and Moduli. The compactification is that given by the
tropical variety of the universal Cox ideal, to be computed as in [45, 47]. The base
space is M0,6 , with points represented by 2 × 6-matrices U as in (1). We encoun-

Fitness, Apprenticeship, and Polynomials

17

tered several themes that are featured in other articles in this book: flag varieties,
Grassmannians, Zn -gradings, Cox rings, Khovanskii bases, and toric ideals. The
connection to spinor varieties was developed in the article [59] with Mauricio Velasco. The formula (2) is derived in [59, Theorem 7.4] for the blow-up of Pn−3 at n
points when n ≤ 8. It is still a conjecture for n ≥ 9. On your trail towards solving
such open problems, fill your backpack with polynomials. They will guide you.
Acknowledgements This article benefited greatly from comments by Lara Bossinger, Fatemeh
Mohammadi, Emre Sertöz, Mauricio Velasco and an anonymous referee. The apprenticeship program at the Fields Institute was supported by the Clay Mathematics Institute. The author also
acknowledges partial support from the Einstein Foundation Berlin, MPI Leipzig, and the US National Science Foundation (DMS-1419018).

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