aa r X i v : . [ m a t h . AG ] D ec Fitness, Apprenticeship, and Polynomials
Bernd Sturmfels
Abstract
This article discusses the design of the Apprenticeship Program at theFields Institute, held 21 August–3 September 2016. Six themes from combinatorialalgebraic geometry were selected for the two weeks: curves, surfaces, Grassmanni-ans, convexity, abelian combinatorics, parameters and moduli. The activities werestructured into fitness, research and scholarship. Combinatorics and concrete com-putations with polynomials (and theta functions) empowers young scholars in alge-braic geometry, and it helps them to connect with the historic roots of their field. Weillustrate our perspective for the threefold obtained by blowing up six points in P . A thematic program on
Combinatorial Algebraic Geometry took place at the FieldsInstitute, Toronto, Canada, during the Fall Semester 2016. The program organizerswere David Cox, Megumi Harada, Diane Maclagan, Gregory Smith, and Ravi Vakil.As part of this semester, the Clay Mathematics Institute funded the “Apprentice-ship Weeks”, held 21 August–3 September 2016. This article discusses the designand mathematical scope of this fortnight. The structured activities took place in themornings and afternoons on Monday, Wednesday, and Friday, as well as the morn-ings on Tuesday and Thursday. The posted schedule was identical for both weeks:MWF 9:00–9:30: Introduction to today’s themeMWF 9:30–11:15: Working on fitness problemsMWF 11:15–12:15: Solutions to fitness problemsMWF 14:00–14:30: Dividing into research teamsMWF 14:30–17:00: Team work on projects
Bernd SturmfelsDepartment of Mathematics, University of California, Berkeley, CA, 94720, United States ofAmerica and Max-Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103Leipzig, Germany, e-mail: [email protected] or [email protected] MWF 17:00–18:00: Teams present findingsTuTh 9:00–12:00: Discussion of the scholarship themeThe term “fitness” is an allusion to physical exercise. In order to improve physicalfitness, many of us go to the gym. A personal trainer can greatly enhance that experi-ence. The trainer develops your exercise plan and he pushes you beyond previouslyperceived limits. The trainer makes you sweat a lot, he ensures that you use exerciseequipment correctly, and he helps you to feel good about yourself afterwards. In thecontext of team sports, the coach plays that role. She works towards the fitness ofthe 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. Tenproblems were posted for each session, and these were available online two or threedays in advance. By design, these demanding problems were open-ended and probeda different aspect of the theme. Section 3 of this article contains the complete list ofproblems, along with a brief discussion and references that contain some solutions.The “apprentices” were about 40 early-career mathematicians, graduate studentsand postdocs, coming from a wide range of backgrounds. An essential feature of theApprenticeship Weeks was the effort to build teams, and to promote collaborationas 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 givento each fitness question. We formed ten teams to work on the problems. At 11:15amwe got together again, and one person from each team gave a brief presentation onwhat had been discussed and discovered. Working on a challenging problem, witha group of new collaborators, for less than two hours created a very intense andstimulating experience. A balanced selection process ensured that each participanthad the opportunity to present for their team at least once.At 2:00pm the entire group re-assembled and they discussed research-orientedproblems for the afternoons. This was conducted in the style of the American In-stitute for Mathematics (AIM), whereby one of the participants serves as the dis-cussion leader, and only that person is allowed to touch the blackboard. This led toan ample supply of excellent questions, some a direct continuation of the morningfitness problems, and others only vaguely inspired by these. Again, groups wereformed for the afternoon, and they engaged in learning and research. Computationsand 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 senseof scholarship among the participants. The morning of these days involved studyingvarious software packages, classical research papers from the 19-th and early 20-th centuries, and the diverse applications of combinatorial algebraic geometry. Theprompts are given in Section 2. The afternoons on discussion days were unstructuredto allow the participants time to ponder, probe, and write up their many new ideas. itness, Apprenticeship, and Polynomials 3
Combinatorial algebraic geometry is a field that, by design, straddles mathemati-cal boundaries. One aim is to study algebraic varieties with special combinatorialfeatures. At its roots, this field is about systems of polynomial equations in severalvariables, and about symmetries and other special structures in their solution sets.Section 5 offers a concrete illustration of this perspective for a system of polyno-mials in 32 variables. The objects of combinatorial algebraic geometry are amenableto 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 mathe-matics. A new field,
Applied Algebraic Geometry , has arisen in the past decade. Thetechniques used there often connect back to 19th and early 20th century work in al-gebraic geometry, which is much more concrete and combinatorial than many recentdevelopments. And, even for her study of current abstract theories, an apprenticemay benefit from knowing the historic origins that have inspired the development ofalgebraic geometry. Understanding these aspects, by getting hands-on experiencesand 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. Thecommon thread can be summarized as: back to the roots. These were given to theparticipants as prompts for explorations and discussions. For several of the partici-pants, 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? Forresearch or for teaching? How often and in which context?Other packages that are useful for our community include
Bertini , PHCpack , , Polymake , Normaliz , GFan . What are these and what do they do? Whodeveloped them and why?Does visualization matter in algebraic geometry?Have you tried software like
Surfex ?Which software tool do you want to learn today?
Bernd Sturmfels
Thursday, August 25: The 19th Century
Algebraic Geometry has a deep and distinguished history that goes back hundredsof years. Combinatorics entered the scene a bit more recently.Young scholars interested in algebraic geometry are strongly encouraged to fa-miliarize themselves with the literature from the 19th century. Dig out papers fromthat period and read them ! Go for the original sources. Some are in English. Do notbe afraid of languages like French, German, Italian.Today we form groups. Each group will explore the life and work of one math-ematician, with focus on what he has done in algebraic geometry. Identify one keypaper written by that author. Then present your findings.Here are some suggestions, listed alphabetically: • Alexander von Brill • Arthur Cayley • Michel Chasles • Luigi Cremona • Georges Halphen • Otto Hesse • Ernst Kummer • Max Noether • Julius Pl¨ucker • 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 2006-07 at the IMA in Minneapolis. Following a suggestion by Doug Arnold (then IMAdirector and SIAM president), it led to the creation of the SIAM activity groupin Algebraic Geometry, and (ultimately) to the SIAM Journal on Applied Algebraand Geometry. The reader is referred to these resources for further information.These interactions with the sciences and engineering have been greatly enhanced bythe 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 aninteraction with that field. Read your paper and then present your findings. Here aresome suggested fields, listed alphabetically: • Approximation Theory itness, Apprenticeship, and Polynomials 5 • 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 19thcentury. Today, we move on to the early 20th century, to mathematics that was pub-lished prior to World War II. You are encouraged to familiarize yourselves withthe literature from the period 1900-1939. Dig out papers from that period and readthem ! Go for the original sources. Some are written in English. Do not be afraid oflanguages like French, German, Italian, Russian.Each group will explore the life and work of one mathematician, with focus onwhat (s)he has done in algebraic geometry during that period. Identify one key paperwritten by that author. Then present your findings.Here are some suggestions, listed alphabetically: • Eugenio Bertini • Guido Castelnuovo • Wei-Liang Chow • Arthur B. Coble • Wolfgang Gr¨obner • 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
Bernd Sturmfels
This section presents the six worksheets for the morning sessions on Mondays,Wednesdays and Fridays. These prompts inspired most of the articles in this vol-ume. 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 P have? Give examples.2. Let f ( x ) = ( x − )( x − )( x − )( x − )( x − )( x − ) and consider the genus2 curve y = f ( x ) . Where is it in the moduli space M ? Compute the Igusainvariants. Draw the Berkovich skeleton for the field of 5-adic numbers.3. The tact invariant of two plane conics is the polynomial of bidegree ( , ) inthe 6 + Bring’s curve lives in a hyperplane in P . It is defined by x i + x i + x i + x i + x i = i = , ,
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 P . The Chow form of X defines ahypersurface in the Grassmannian Gr ( , P ) . Points are lines that meet X . Findthe dimension and (bi)degree of its singular locus.6. What are the equations of the secant varieties of elliptic normal curves ?7. Let X P be the toric variety defined by a 3-dimensional lattice polytope, as inMilena Hering’s July 18-22 course. Intersect X P with two general hyperplanes toget 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 , n .Write these equations in Macaulay2 format for n = n =
6. Can you seethe y -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 × { x + y + z = } . Howcan you recover the curve from that matrix?10. The moduli space M of genus 7 curves has dimension 18. What is the codimen-sion of the locus of plane curves? Hint: Singularities are allowed. Wednesday, August 24: Surfaces
1. A nondegenerate surface in P n has degree at least n −
1. Prove this fact and de-termine all surfaces of degree n −
1. Give their equations.2. How many lines lie on a surface obtained by intersecting two quadratic hyper-surfaces in P ? Find an instance where all lines are defined over Q .3. What is the maximum number of singular points on an irreducible quartic surfacein P ? Find a surface and compute its projective dual . itness, Apprenticeship, and Polynomials 7
4. Given a general surface of degree d in P , the set of its bitangent lines is a surfacein Gr ( , P ) . Determine the cohomology class (or bidegree) of that surface.5. Pick two random circles C and C in R . Compute their Minkowski sum C + C and their Hadamard product C ⋆ C . 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 manytorus-fixed points are there on the Hilbert scheme of 20 points in P ? What canyou say about the graph that connects them?9. State the Hodge Index Theorem . Verify this theorem for cubic surfaces in P , byexplicitly 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 ( , ) with precisely 16 non-zero Pl¨ucker coordinates. As inJune Huh’s July 18-22 course, determine the Chow ring of its matroid .2. The coordinate ring of the Grassmannian Gr ( , ) is a cluster algebra of finitetype. What are the cluster variables? List all the clusters.3. Consider two general surfaces in P whose degrees are d and e respectively. Howmany lines in P 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 ( ) is a six-dimensional subvariety of P × P × P . Compute its ideal and determine its tropicalization.6. Classify all toric ideals that arises as initial ideals for the flag variety above. Foreach such toric degeneration, compute the Newton-Okounkov body .7. The Grassmannian Gr ( , ) has dimension 12. Four Schubert cycles of codimen-sion 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-dimensionalversions of the Grassmannian. How are they related?9. The coordinate ring of the Grassmannian Gr ( , ) is Z -graded. Determine theHilbert series and the multidegree of Gr ( , ) for this grading.10. The Lagrangian Grassmannian parametrizes n -dimensional isotropic subspacesin C n . Find a Gr¨obner 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 inSym ( R ) ≃ R . Determine the face poset of this convex cone. Bernd Sturmfels
2. Consider smooth projective toric fourfolds with eight invariant divisors. What isthe maximal number of torus-fixed points of any such variety?3. Choose three general ellipsoids in R 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 de-rived from the Hodge Index Theorem (for algebraic surfaces).5. The blow-up of P at six general points is a threefold that contains 32 specialsurfaces (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 × × ( , ) in the unitsphere in R . Describe its convex hull. Hint: Calibrations, Orbitopes.10. Examine Minkowski sums of three tetrahedra in R . What is the maximum num-ber of vertices such a polytope can have? How to generalize? Wednesday, August 31: Abelian Combinatorics
1. The intersection of two quadratic surfaces in P is an elliptic curve . Explain itsgroup structure in terms of geometric operations in P .2. A 2006 paper by Keiichi Gunji gives explicit equations for all abelian surfaces in P . 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 q ( t ; z ) . Verify the functional equation.4. Theta functions with characteristics q [ e , e ′ ]( t ; z ) are indexed by two binary vec-tors e , e ′ ∈ { , } g . They are odd or even. How many each?5. Fix the symplectic form h x , y i = x y + x y + x y + x y + x y + x y on the64-element vector space ( F ) . Determine all isotropic subspaces.6. Explain the combinatorics of the root system of type E . How would you choosecoordinates? 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 R defines a degeneration of abelian 5-folds. Describe itsperiodic tiling on R 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 Q by itness, Apprenticeship, and Polynomials 9 x + x y + x z + x y + x yz − x z + xy + xy z − xyz + xz + y − y z − y z − yz − z Compute its limit in
Alexeev’s moduli space for the 2-adic valuation.10. Let Q be the theta divisor on an abelian threefold X . Find n = dim H ( X , k Q ) .What is the smallest integer k such that k Q is very ample? Can you compute (in Macaulay2 ) the ideal of the corresponding embedding X ֒ → P n − ? 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 ) = ( b ) =
4, deg ( c ) =
5, and deg ( d ) =
9. Classify all homogeneous ideals I suchthat S / I has Hilbert function identically equal to 1.3. Consider the Hilbert scheme of eight points in affine 4-space A . Identify a pointthat is not in the main component. List its ideal generators.4. Let X be the set of all symmetric 4 × R × that have an eigenvalueof multiplicity ≥
2. Compute the C -Zariski closure of X .5. Which cubic surfaces in P are stable? Which ones are semi-stable?6. In his second lecture on August 15, Valery Alexeev used six lines in P to con-struct a certain moduli space of K3 surfaces with 15 singular points. List the mostdegenerate points in the boundary of that space.7. Find the most singular point on the Hilbert scheme of 16 points in A .8. The polynomial ring C [ x , y ] is graded by the 2-element group Z / Z wheredeg ( x ) = ( y ) =
1. Classify all Hilbert functions of homogeneous ideals.9. Consider all threefolds obtained by blowing up six general points in P . Describetheir 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 dimension12. Determine the number of i -faces for i = , , , . . . , Solutions to several of the sixty fitness problems can be found in the 16 articles ofthis volume. The articles are listed as the first 16 entries in our References. Theywill be published in the order in which they are cited in this section. In what followswe also offer references for other problems that did not lead to articles in this book.
Notes on Curves
1. Castelnuovo classified the degree and genus pairs ( d , g ) for all smooth curves in P n . This was extended to characteristic p by Ciliberto [25]. For n = , d =
6, thepossible genera are g = , , , ,
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 approachusing 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 forbitangents 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 × X P is the volume of its lattice polytope P .The genus of a complete intersection in X P was derived by Khovanskii in 1978. Werecommend the tropical perspective offered by Steffens and Theobald in [53, § n = § M 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 beautifulintroduction 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 P into P via a Q -basis for the cubics that vanish at five rational points in P .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 in16 variables modulo an ideal generated by 20 quadrics. Ideal generators that areuniversal over the base M , are listed in [47, Proposition 2.1]. Ideals of points onthe surface are torus translates of the toric ideal of the 5-dimensional demicube D .For six points in P we refer to Bernal, Corey, Donten-Bury, Fujita and Merz [7].7. This is the clique complex of the Schl¨afli graph . The f-vector of this simplicialcomplex is ( , , , , , ) . The Schl¨afli graph is the edge graph of the E -polytope , denoted 2 , which is a cross section of the Mori cone of the surface.8. The torus-fixed points on Hilb ( P ) are indexed by ordered triples of partitions itness, Apprenticeship, and Polynomials 11 ( l , l , l ) with | l | + | l | + | l | =
20. The number of such triples equals 341 , ( , r − ) where r is the rank of thePicard group. This is r = Notes on Grassmannians
1. See the article by Wiltshire-Gordon, Woo and Zajackowska [9].2. In addition to the 20 Pl¨ucker coordinates p i jk , one needs two more functions,namely p p − p p and p p − p p . The six boundary Pl¨ucker co-ordinates p , p , p , p , p , p are frozen. The other 16 coordinates arethe cluster variables for Gr ( , ) . 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 ( , ) fourtimes. For this problem, and many other Schubert problems, instances exist whereall 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 arestudied, respectively, in integrable systems and in geometric representation theory .9. A formula for the Z n -graded Hilbert series of Gr ( , n ) is given by Witaszek [63, § § Macaulay2 com-mands
Grassmannian and multidegree . Escobar and Knutson [12] deter-mine the multidegree of a variety that is important in computer vision.10. The coordinate ring of the Lagrangian Grassmannian is an algebra with straight-ening law over a doset . This stands for double poset. See the exposition in [48, § Notes on Convexity
1. The face lattice of the cone of non-negative binary forms of degree d is describedin 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 resolvedby 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 Con-vex Hull of Ellipsoids by Nicola Geismann, Michael Hemmer, and Elmar Sch¨omer.4. We refer to Ewald’s textbook, specifically [28, § IV.5 and § VII.6].
5. The relevant polytope is the 6-dimensional demicube; its 32 vertices correspondto 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 isomor-phism. 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 thereare 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 identi-cal regular tetrahedra by removing the four pairs of corresponding vertices. It is theconvex hull of the points e i ⊕ e j in R ⊕ R where i , j ∈ { , , , } with i = j . Usingthe software Polymake , we find its f-vector to be ( , , , , , ) .9. The faces of the Grassmann orbitopes conv ( Gr ( , n )) for n ≥ n = et al. in[35, § Notes on Abelian Combinatorics
1. A beautiful solution was written up by Qiaochu Yuan when he was a high schoolstudent; see [62]. The idea is to simultaneously diagonalize the two quadrics, thenproject 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 2 g pairs ( e , e ′ ) , we check whether e · e ′ is even or odd. There are2 g − ( g + ) even theta characteristics and 2 g − ( g − ) odd theta characteristics.5. The number of isotropic subspaces of ( F ) is 63 of dimension 1, it is 315 indimension 2, and it is 135 in dimension 3. The latter are the Lagrangians [46, § E has 63 positive roots. They are discussed in [46, § E with the 63 non-zerovectors in ( F ) . Two roots have inner product zero if and only if the correspondingvectors in ( F ) \{ } are orthogonal in the setting of Problem 5. See [46, Table 1].8. This refers to Gwena’s article [31]. Since the matroid R is not co-graphic, thecorresponding 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 k Q is very ample for k =
3. This embeds any abelian threefold into P . For products of three cubic curves, each in P , this gives the Segre embedding. itness, 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
Aron-hold invariant and it vanishes when the ternary cubic is a sum of three cubes oflinear forms. Both invariants vanish when the cubic curve has a cusp.2. This refers to extra irreducible components in toric Hilbert schemes [42]. Theseschemes were first introduced by Arnold [19], who coined the term
A-graded al-gebras . Theorem 10.4 in [54] established the existence of an extra component for A = ( ) . 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 A hastwo irreducible components. An explicit point in the non-smoothable component isgiven 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 workover the real numbers R . The analogous set over C is the hypersurface of a sum-of-squares 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, § P . The precise space depends on a choice of parameters [18, § ( , ) , so the most degenerate points correspond to theseven 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 andeven boxes in its Young diagram. The resulting Hilbert functions h : Z / Z → N are ( h ( even ) , h ( odd )) = ( k + m , k ( k + ) + m ) or (( k + ) + m , k ( k + ) + m ) , where k , m ∈ N . This was contributed by Dori Bejleri. For more details see [21, § P n − at n points is a Mori dream space. Its Cox ring has 2 n − generators, constructed explicitly by Castravet and Tevelev in [23]. These form aKhovanskii 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 bundleon P with n marked points. The relevant moduli space is M , n .10. The moduli space of tropical curves of genus 5 serves as the first example inthe 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. The author of this article holds the firm belief that algebraic geometry concerns thestudy of solution sets to systems of polynomial equations. Historically, geometersexplored curves and surfaces that are zero sets of polynomials. It is the insightsgained from these basic figures that have led, over the course of centuries, to theprofound depth and remarkable breadth of contemporary algebraic geometry. How-ever, many of the current theories are now far removed from explicit varieties, andpolynomials are nowhere in sight. What we are advocating is for algebraic geom-etry to take an outward-looking perspective. Our readers should be aware of thewealth of applications in the sciences and engineering, and be open to a “back tothe basics” approach in both teaching and scholarship. From this perspective, theinteraction with combinatorics can be particularly valuable. Indeed, combinatoricsis known to some as the “nanotechnology of mathematics”. It is all about explicitobjects, 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 { , , , , , } whose cardinality is odd: p , p , . . . , p , p , p , p , . . . , p , p , p , p , . . . , p . The polynomial ring Q [ p ] is Z -graded by setting degree ( p s ) = e + (cid:229) i ∈ s e i , where e , e , . . . , e is the standard basis of Z . Let X be a 5 × I be the kernel of the ring map Q [ p ] → Q [ X ] that takes the variables p s to the deter-minant of the submatrix of X with column indices s and row indices 1 , , . . . , | s | .The ideal I is prime and Z -graded. It has multiple geometric interpretations.First of all, it describes the partial flag variety of points in 2-planes in hyperplanesin P . This flag variety lives in P × P × P , thanks to the Pl¨ucker embedding. Itsprojection into the factor P is the Grassmannian Gr ( , ) of 2-planes in P . Flagvarieties 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 minimalquadratic generators. Sixty generators are unique up to scaling in their degree:degree ideal generator ( , , , , , , ) p p − p p + p p − p p ( , , , , , , ) p p − p p + p p − p p · · · · · · · · · · · · ( , , , , , , ) p p − p p + p p − p p ( , , , , , , ) p p − p p + p p · · · · · · · · · · · · ( , , , , , , ) p p − p p + p p ( , , , , , , ) p p − p p + p p − p p · · · · · · · · · · · · ( , , , , , , ) p p − p p + p p − p p itness, Apprenticeship, and Polynomials 15 The other six minimal generators live in degree ( , , , , , , ) . These are the 4-term Grassmann-P¨ucker relations, like p p − p p + p p − p p .Here is an alternate interpretation of the ideal I . It defines a variety of dimension15 = (cid:0) (cid:1) in P known as the spinor variety . In this guise, I encodes the algebraicrelations among the principal subpfaffians of a skew-symmetric 6 × { , , , , , } of even cardinality. Thetrick is to fix a natural bijection between even and odd subsets. This variety is similarto the Lagrangian Grassmannian seen in fitness problem 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 Khovanskiibases for Q [ p ] / I and hence toric degenerations of the spinor variety in P . Theircombinatorics is recorded in a list of Newton-Okounkov polytopes with 32 vertices.Each of these polytopes comes with a linear projection to the 6-dimensionaldemicube, which is the convex hull in R of the 32 points deg ( p s ) . We saw thisdemicube in fitness problem Cox rings , and their Khovanskii bases, similarto those in the article by Bernal, Corey, Donten-Bury, Fujita and Merz. We begin byreplacing the generic 5 × X by one that has the special form in [23, (1.2)]: X = u x u x u x u x u x u x u y u y u y u y u y u y u v x u v x u v x u v x u v x u v x v y v y v y v y v y v y v x v x v x v x v x v x . Now, the polynomial ring Q [ X ] gets replaced by k [ x , x , . . . , x , y , y , . . . , y ] where k is the field extension of Q generated by the entries of a 2 × U = (cid:18) u u u u u u v v v v v v (cid:19) . (1)We assume that the 2 × U are non-zero. Let J denote the kernel of theodd-minors map k [ p ] → k [ X ] as before. The ideal J is also Z -graded and it strictlycontains the ideal I . Castravet and Tevelev [23, Theorem 1.1] proved that k [ p ] / J isthe Cox ring of the blow-up of P k 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 Z fur-nishes the grading. The affine variety in A 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 P k is problem J = I + u ∗ I . (2)Here u is a vector in ( K ∗ ) that is derived from U . The ideal u ∗ I is obtained from I by scaling the variables f s with the coordinates of u . In particular, the Cox ideal J isminimally generated by 132 quadrics. Now, there are two generators in each of thesixty Z -degrees in our table, and there are 12 generators in degree ( , , , , , , ) .Following [60, Example 7.6], we fix the rational function field k = Q ( t ) and set U = (cid:18) t t t t t t t t t t (cid:19) . The ring map k [ p ] → k [ X ] now maps the variables p s like this: p x p x y x t − ( x x y + y x x ) t + ( y x x + x x y ) t − x y x t p x y x y x t − ( y x x y x + x y x x y + · · · + x x y y x ) t + · · · Here is a typical example of a Z -degree with two minimal ideal generators: ( , , , , , , ) p p − p p + p p − p p ( , , , , , , ) t p p − t p p + t p p + p p The algebra generators p s 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 theunderlined monomials. Its toric ideal in ( J ) is generated by 132 binomial quadrics:degree pair of binomial generators for in ( J )( , , , , , , ) p p − p p p p − p p ( , , , , , , ) p p − p p p p − p p · · · · · · · · · ( , , , , , , ) p p − p p p p − p p ( , , , , , , ) p p − p p p p − p p · · · · · · · · · ( , , , , , , ) p p − p p p p − p p ( , , , , , , ) p p − p p p p − p p · · · · · · · · · ( , , , , , , ) p p − p p p p − p p These 132 binomials define a toric variety that is a degeneration of our universaltorsor. 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 inconformal field theory [39]. Beautiful polynomials can bring the sciences together.Let us turn to another fitness problem. The past three pages offered a capoeiraapproach to M , , with points represented by 2 × U as in (1). We encoun- itness, Apprenticeship, and Polynomials 17 tered several themes that are featured in other articles in this book: flag varieties,Grassmannians, Z n -gradings, Cox rings, Khovanskii bases, and toric ideals. Theconnection to spinor varieties was developed in the article [59] with Mauricio Ve-lasco. The formula (2) is derived in [59, Theorem 7.4] for the blow-up of P n − at n points when n ≤
8. It is still a conjecture for n ≥
9. On your trail towards solvingsuch open problems, fill your backpack with polynomials. They will guide you.
Acknowledgements
This article benefited greatly from comments by Lara Bossinger, FatemehMohammadi, Emre Sert¨oz, Mauricio Velasco and an anonymous referee. The apprenticeship pro-gram at the Fields Institute was supported by the Clay Mathematics Institute. The author alsoacknowledges partial support from the Einstein Foundation Berlin, MPI Leipzig, and the US Na-tional Science Foundation (DMS-1419018).
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