# Udine Examples. Alessio Corti, Imperial College London. London, 23rd Sep Abstract

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1 Udine Examples Alessio Corti, Imperial College London London, rd Sep 014 Abstract This is a list of exercises to go with the EMS School New perspectives on the classification of Fano manifolds in Udine, Sep 9 Oct 0, Contents 1 Log del Pezzo surfaces Fano polygons, polytopes and mutations Toric complete intersections 4 4 Picard Fuchs differential equations 4 5 Quantum cohomology 5 6 Conjecture B 1 7 Appendix 14 General advice Realistically, there is no way you can do all of these problems during the School. You should choose some that you like and get help setting yourself up. Help is available by the following people on the following topics: Mohammad Akhtar Mutations; Liana Heuberger Classification of the 9 (6) surfaces, toric complete intersections; Alessandro Oneto Quantum orbifold cohomology; Andrea Petracci Quantum orbifold cohomology; 1 This sheet was put together in great haste and I expect that it contains several errors and misprints. I keep a corrected and updated version of this document on my teaching page acorti/teaching.html 1

2 Thomas Prince Mutations, toric complete intersections; Picard Fuchs operators; Ketil Tveiten Rigid maximally mutable Laurent polynomials. Topology of f(x, y) = 0. Monodromy and ramification index. 1 Log del Pezzo surfaces (1) Let X be the stack [(xy + z w = 0)/µ r ] where µ r acts with weights (1, wa 1, a) and hcf(r, a) = 1 and hcf(wr, wa 1) = 1 (so the moduli space X is the surface quotient singularity 1/n(1, p 1) with n = wr, p = wa). Show that Ext 1 O X (Ω 1 X, O X) = C m 1 where w = mr + w 0, 0 w 0 < r. () Let the isolated surface quotient singularity X = 1/n(1, p 1) have Gorenstein 1 index r =. Show that: either X is of class T ; or X = (m+) (1, m + 1) with content { m, 1 6 (1, 1)} 1 ; or X = (m+1) (1, 6m + 1) with content { m, 1 (1, 1)}. Draw some of the cones and stare at them. Do a similar analysis for Gorenstein index, 4 and 5. () Consider a del Pezzo surface X with k quotient singularities 1/(1, 1). (i) Somehow or other convince yourself that h 0 n(n + 1) (X, nk) = 1 + K 1 if n 1 (mod ) + k 0 if n 0, (mod ) (the official reference of this kind of thing is Reid s Young Person s Guide.) (ii) Show that K X = 1 n 5k where n Z is the topological Euler number e(x0 ) of the smooth locus of X, i.e. the complement X \ S of the set S of singular points of X. (iii) Assume that k = 1 and X has no free ( 1)-curves (that is, ( 1)-curves contained in the smooth locus of X). Show that X = P(1, 1, ). (Hint: use the minimal model program.) (iv) Conclude that if k = 1 then X is the blow up of P(1, 1, ) at 8 general points. (v) (Harder) Study the case k = in a similar vein. (4) Procure yourself 5 general lines in P, blow up the 10 points of intersection to obtain a surface Y, contract the strict transforms of the 5 general lines to obtain a surface X. Show that X is a Del Pezzo surface with 5 singular points 1/(1, 1) and degree K = /. Show that X has a free ( 1)-curve, that is a ( 1)-curve not intersecting any of the singular points. Contracting this ( 1)-curve yields a surface X 1 with 5 singular points 1/(1, 1) and degree K X = /. Show that X 1 in turn has no free ( 1)-curves. Give an alternative Italian-style birational geometry construction of X 1. (I don t know how to do this.)

3 Fano polygons, polytopes and mutations (5) Find 16 reflexive Fano polygons. (DON T ask google) and find all the maximally mutable Laurent polynomials supported on them. (6) Find an example of an immutable Fano polygon. [Try P(, 5, 11). Now find another one.] (7) A polygon P is called centrally symmetric if v P implies that v P. Show that any centrally symmetric polygon is minimal. It is conjectured that there exists at most one centrally symmetric polygon in each mutation equivalence class can you prove this? (8) Let P R be a Fano polygon. Let v 1,..., v k be the vertices of P in counterclockwise order, and write v i v i 1 = m i e i where m i N and e i is a primitive lattice vector. Show that the Minkowski factors of P correspond to integer linear combinations: wi e i = 0, where 0 w i < m i Generalize to higher dimensional Minkowski decompositions. (9) Let P R be a Fano polytope in D. Show that there are finitely mutations out of P. (10) In this question you will learn how to encode lattice polygons by using continued fractions representing zero. (i) For integers a i define [a 0,..., a n ] = a 0 a 1 a a 1 a n The purpose of this part of the question is to make sense of the expression [a 0,..., a n ] for a i Z. Indeed write: ( q n 1 p n 1 q n p n ) ( ) ( ) = 1 a 0 1 a m and define [a 0,..., a n ] = pn q n. Show that this equals the above definition when all a i. (ii) Let P be a Fano polygon. Construct a subdivision of the spanning fan of P corresponding to the minimal resolution of the toric surface X P. Starting from a

4 vertex v = e 0 of P, let e 0, e 1,..., e n be the generators of the rays of the subdivision in counterclockwise order. Show that ( ) e i e i+1 ( ) ( 0 1 = 1 where a i = Ei is the selfintersection of the corresponding divisor in the minimal resolution. (iii) Show that [a 0,..., a n ] = 0 with multiplicity one. Viceversa, starting with a sequence a 0,..., a n of integers such that [a 0,..., a n ] = 0 with multiplicity one, construct a corresponding Fano polygon. a i e i 1 e i ) (11) Write down and understand the complete mutation graph of the polygon with vertex matrix: ( (we don t know how to do this. If you do, let us know.) ) Toric complete intersections (1) Find all singularities of X and compute KX for X one of the following toric complete intersections: (i) (ii) (iii) (iv) Picard Fuchs differential equations (1) Consider the good-old complete elliptic integral π(λ) = 1 d λ π i x(x 1)(x λ) 4

5 (i) By using the residue theorem around λ = 0, show the power series expansion ( ) 1/ λ n π(λ) = n 0 n (ii) Conclude that π(λ) is a solution of the Picard Fuchs ordinary differential equation: (λ(λ 1) d d + (λ 1) d λ dλ + 1 ) π(λ) = 0 4 [For an alternative approach see Clemens A scrapbook of complex curve theory.] (14) (i) Choose a few (say 5) polygons from Figure 4 (make sure you choose them with different values of n); (ii) For each of these polygons P figure out all of the maximally mutable Laurent polynomials f(x, y) with Newt f = P ; (iii) Compute the generic genus of the completion and normalization of the curve f(x, y) = 0; (iv) Can you now guess the answer to the following question: If P N R is a Fano polygon and f(x, y) a generic maximally mutable Laurent polynomial with Newt f = P, what is the genus of the completion normalization of the curve f(x, y) = 0? Can you prove it? 5 Quantum cohomology In questions 15 and 16, you are asked to compute part of the small quantum cohomology of Del Pezzo surfaces of degree and naïvely from the definition. (15) Cubic surface (i) Let X = X P be a nonsingular cubic surface. Let A be the class of a hyperplane section and consider the subspace of H X with basis 1, A, A = pt. Show that quantum multiplication by A preserves this subspace and it is given by the matrix 0 108q 756q M = 1 9q 108q [Hint. The relevant enumerative information is: A, A, A 1 = 7, A, A, A = 1 pt = 1 7, A, A, A = 7 pt, pt = 7 84.] (ii) This is one of the simples examples of a mirror theorem: e 6q ψ 0 cf. Q 8(4) satisfies the hypergeometric operator D q(d + 1)(D + ). 5

6 (16) Degree two del Pezzo X = X 4 P(1, ) Make a similar discussion for the del Pezzo surface of degree, X = X 4 P(1, ): (i) In the obvious basis 0 55q 7, 488q M = 1 8q 55q (For instance, 7, 488 = 6 148, where 148 is the number of cubics through two general points of X.) (ii) Make contact with the appropriate hypergeometric differential operator D 4q(4D + 1)(4D + ). The next 7 questions form a series on the basics of orbi-curves, which should help to later build a feeling for how quantum orbifold cohomology is put together. An orbicurve is a nodal twisted curve ( C, x i (r i ) ) where all points with non-trivial stabiliser are marked with an isomorphism G xi = µ ri (sometimes I omit the marked points from the notation). (17) (i) Persuade yourselves that the orbifold fundamental group of a smooth orbicurve ( C, x i (r i ) ) is π orb 1 C = π 1 ( C {xi } ) / γ r i i where γ i are small loops around the punctures. (ii) Let C be a smooth orbi-curve and G a finite group. Show that to give a representable morphism C BG is equivalent to give a group homomorphism π orb 1 C G which sends each γ i to an element of order r i. The data is also equivalent to give a principal G-bundle on C, that is a space π : G E C (where C is the coarse moduli space of C) which is a principal G-bundle over C {x i } and has inertia group µ ri above x i. (18) Show Riemann-Roch and Serre duality for an orbi-curve C. For example, if L is a line bundle, then we get representations of µ ri on the fibre L xi of L at x i and a Riemann-Roch formula χ(c, L) = deg L + 1 g k i r i (19) If ( C; x i (r i ) ) is a n-pointed orbi-curve and f : ( C; x i (r i ) ) X is a stable representable morphism, then f T X makes sense is an orbi-bundle and µ ri = G xi acts through the representation into G f(x) ; we label the representation at x i by its weights 6

7 0 w i,j < r i ; persuade yourself that the expected dimension of the moduli space is dim X 0,n,β = χ ( C, f T X ) + n = = K X β + dim X + n n i=1 dim X j=1 w i,j r i [Hint. Denote by L f = f Ω 1 X Ω1 C the cotangent complex of the morphism f. The deformation theory of f is controlled by the hyperext algebra Ext ( L f, O C ).] (0) Let f : ( C, x i (r i ) ) X be a stable morphism. Let us assume that f is a embedding locally at every point of C. In this case, there is a locally free sheaf N f, the normal bundle of f, defined by the sequence 0 Nf Ω1 X Ω1 C 0 This needs to be taken with a pinch of salt: show that, when C has nodes, the natural sheaf homomorphism T X N f is not surjective. (1) Find concrete models for the moduli stacks of stable morphisms of degree from orbi-curves to P(1, 1, ). Determine which components have the correct dimension, which are smooth as Deligne-Mumford stacks, and the nature of all the singular points. [Degree two is very tough, but try to do at least the case of morphisms of degree /.] () Give a sensible definition of a stacky topological Euler number of a smooth stack curve. State some properties of the topological Euler number. Let C be a smooth proper curve and G a finite group acting on C: calculate the stacky topological Euler number of the stack [C/G] in terms of vertices, edges and faces of a G-invariant cellular decomposition of C. () (i) Given a Deligne-Mumford stack X, build a model for the simplicial stack made of moduli stacks X 0,,0 of genus 0 -pointed stable morphisms of degree 0 in terms of higher inertia stacks X. Carefully identify all degeneneracy and face maps. (ii) Build a model for the moduli stack X 1,1,0 of genus 1 1-pointed stable morphisms of degree 0. Be careful: this is rather tricky. For instance if X = BG, then X 1,1,0 is a moduli stack of G-twisted covers. In the next two questions, you are asked to compute the small quantum orbifold cohomology of a simple explicit stack X naïvely from the definition. This is hard work but it does give a body to a very abstract formalism. 7

8 If w = (w 0,..., w n ) is an integer vector and X = P w the corresponding weighted projective space, than the components of the inertia stack are in 1-to-1 correspondence with the set F = { ki w i i = 0,..., n; 0 k i < w i } We denote by X 0,n,d (f 1,..., f n ) the connected component of X 0,n,d of stable morphisms which evaluate in the components of inertia corresponding to f 1,..., f n F. I often confuse degree in cohomology with degree in the Chow ring please sort out the factors of on your own. (4) X = P(1, 1, ) (i) Show that H orb X is generated as a vector space by classes 1, η 1, A = O(1), η, A in cohomology degrees 0, /, 1, 4/,. (ii) Show directly from the definition that η 1 η 1 = η, and η 1 η = A = 1 pt. [Hint. For the first one look for constant representable morphisms in X 0,,0 ( 1 ). Note that the last point evaluates with inversion in P(). The moduli space has virtual dimension 0 + / / /=0; etc. For the second look for constant representable morphisms in X 0,,0 (1/, /, 0); the expected dimension of the moduli space is 0+ / 4/ = 0; the relevant component of the moduli space is isomorphic to P(); by definition η 1 η whatever it is, it has degree P() 1 = 1/.] Finally, it is clear that η η 4/ 4/ 4/ < 0. = e 1 = 1 pt = 0; indeed, X 0,,0 ( ) has virtual dimension 0 + (iii) First note that codim q = = 5/ (why?). In the basis above, write down the matrix of quantum multiplication by A: 0 aq bq 1 0 M = cq and show that a = b = c = 1/ by interpreting the unknown entries a, b, c in terms of stacky Gromov-Witten invariants. [Hints: First, from A η 1 = 1aq and integrating against A : 1/ aq = deg A aq = 1, A aq = A η 1, A = A, η 1, A 1/ q 8

9 or a = e 1(A) e (η 1 ) e (A ) e(e). X 0,,1/ Here X 0,,1/ parameterises representable morphisms with image a curve of degree 1/ on X; knowing what these curves are, we must be looking at X 0,,1/ (0, 1/, 0); the virtual dimension is dim X 0,,1/ (0, 1/, 0) = 5/ + / =. The virtual dimension is the actual dimension and the problem is unobstructed; you can integrate: a = e 1(A) e (η 1 ) e (A ) = X 0,,1/ Sanity check: a = A, η 1 = e 1(A) e (η 1 ) = A = 1 X 0,,1/ P(1,), A 1/ = A, η 1, A 1/ = (by the divisor axiom) =, pt 1/ = 1/: there is just one orbi-line of degree 1/ that passes through the 1/ η 1 singular point and one additional general point. The corresponding stable morphism has no automorphisms, hence this line contributes with multiplicity 1 to η 1, pt 1/. Second, from A η = bq η 1, we derive 1 b = b η 1, η = A, η, η 1/ The relevant moduli space is X 0,,1/ (1, /, /); it has expected dimension 5/ + 4/ 4/ = 1. The only way to achieve this is by gluing a morphism in X 0,,0 ( ) with one in X0,,1/ (1/, 0); the two orbi-curves glue as a nontrivial twisted curve and the map is constant on the first component. This space is two dimensional; we have to deal with a one-dimensional x x x 1 / / / 1/ ( Figure 1: X 0,, 1 0,, ) obstruction bundle. Note that the first component is in X 0,,0 ( ) which has negative 9

10 virtual dimension 0 + (4/) = but it is still there. calculate b using the associativity relations: Fortunately, we can A η = A (A η ) = bqa η 1 = abq 1, hence abq = A η, pt = A, η, pt q. We calculate an integral over X 0,,/ (0, /, 0); the generic point of this moduli space is a stable morphism from a reducible curve with three components: The key thing to keep x 1 x x 1/ 1/ / / / ( Figure : X 0,, 0,, 0) in mind is that the corresponding morphism always has a µ of automorphisms over P(1, 1, ), coming from the central component C on which the morphism is constant. The central component maps to Bµ and it carries an induced µ -bundle; this bundle has a µ of automorphisms which survive as nontrivial automorphisms of C over Bµ. Having said this, we can now calculate b: ab = 1 b = 1 e 1(pt) e (η ) e (pt) = 1 1 = 1 X 0,,/ (0,/,0) P() 9 that is, b = 1/. Third, show that c = a. Indeed, from A A = cq η, we get 1/ cq = cq deg η η 1 = cq η, η 1 = A, A, η 1 1/ q and c = A, A, η 1 1/ = A, pt, η 1 1/ = 1/ pt, η 1 1/ = 1 as before.] (iv) Let D = q d dq and consider the quantum differential equation DΨ = ΨM for Ψ: C End H orb (X, C). In the given basis, write Ψ = (ψ 0,..., ψ n ) where ψ i are column vectors; find the ordinary differential equation satisfied by ψ 0. [Hint. (D /)(D 1/)D ψ 0 = (D /)(D 1/)D ψ = (D /)(D 1/)Dψ 4 = (D /)(D 1/)q 1/ ψ = q 1/ (D 1/)Dψ = q 1/ (D 1/)q 1/ ψ 1 = q / Dψ 1 = qψ 0 ] 10

11 (5) X = X P(1, ) (i) Note that X can be written as ( yx0 + a 1 (x, x ) = 0 ) P(1, ) Build a mental picture of X by studying the obvious birational map X P : X is obtained by blowing up three collinear points and contracting the proper transform of the line (x 0 = 0). Let A = O X (1). In particular, on X, there are: Three lines of A-degree 1/; Three fibrations by conics of A-degree 1; One map to P. (ii) Convince yourself that the orbifold cohomology of X has basis 1, A, η, A in degrees 0, 1, 1, ; deg A = / and deg η = 1/. (iii) Use the information in (i) to show that quantum multiplication by A is given in this basis by the following matrix where codim q = : 0 q A, A, pt M = q 1 A, η, A 1 q A, pt, A 1 0 q A, A, η = q = 1 0 q 1 q 0 q (iv) Verify directly that the cyclic vector ψ 0 cf. Q 8(4) satisfies the expected hypergeometric operator D (D 1) q(d + 1)(D + ). [Hint: First rewrite Dψ = ψm in the new basis φ 0 = ψ 0 φ 1 = Dψ 0 = ψ 1 φ = ψ + q 1 ψ φ = ψ A small calculation shows that the equation in the new basis is: 0 q DΦ = Φ q 1 q q

12 In this form it is easy to calculate the equation satisfied by φ 0 = ψ 0.] (v) Show that the quantum products calculated in (iii), together with associativity, determine the whole small quantum cohomology ring. determines the curious Gromov-Witten number: η, η, η 1 = 4. In particular, show that this (vi) The direct calculation of the number in (v) leads to a beautiful case study in excess intersection theory: the expected dimension of the moduli space is 1+ 1 = 0; x 1 1/ 1/ x x 1/ 1/ 1/ ( Figure : X 1 ) 0,,1/ however, the picture shows an actual moduli space of dimension 1 (the four points on the component on which the morphism to X is constant). If you feel brave enough, calculate η, η, η 1 = 4 by a study of the virtual class. 6 Conjecture B (6) Verify Conjecture B for the following surfaces taken from the 6 surfaces in Table : (i) No. 6, extended to all of H < orb (X, C) C( K X). (ii) No., nonextended, with mirror f(x, y) = (1 + y) xy + 1 y + xy (iii) No. 15, nonextended, with mirror f(x, y) = (1 + x) (1 + y) xy (iv) No. 4, nonextended, with mirror + (1 + y) x + y 4 f(x, y) = ( x + 1 ) 4 ( y y x + y ) 4 x 1

13 (7) In this question we verify Conjectures A and B in a simple -fold situation: (i) Find all the rigid maximally mutable Laurent polynomials on the D Fano polytope with vertex matrix: You should find precisely two such polynomials: f(x, y, z) = xz 1 + yz 1 + x 1 y 1 z 1 + x y z 1 + x 1 y z 1 + y 1 z 1 + z + z 1 g(x, y, z) = xz 1 + yz 1 + x 1 y 1 z 1 + x y z 1 + x 1 y z 1 + y 1 z 1 + z + z 1 (ii) Compute the first few terms of the quantum period and the Picard Fuchs differential operators for these two polynomials. (Don t be too proud: go and get some help: ask someone with a computer): L f = D + 4t (D + 1)(7D + 14D + 8) + t 4 18(D + 1)(D + )(D + ) L g = D + 8t (D + 1)(5D + 10D + 6) t 4 144(D + 1)(D + )(D + ) (iii) Show that Vol(P ) = 48. Go on a table of Fano -folds and find that there are two families with K = 48. Show that f is mirror to X (1,1) P P ; and g is mirror to P 1 P 1 P 1. (8) Find all the rigid maximally mutable Laurent polynomials on the 4D Fano polytope with vertex matrix: You should just find f(x, y, z, w) = x + y + z + w + xyz + 1 x y z w (ii) Compute the first few terms of the quantum period and the Picard Fuchs differential operator: D t 4 (D + 1)(D + ) (D + ) (iii) Show that f is mirror to the 4-dimensional quadric. 1

14 (9) Find all the rigid maximally mutable Laurent polynomials on the 4D Fano polytope with vertex matrix: You should just find f(x, y, z, w) = x + y + z + w + z xyw + 1 xy + 1 xyw + 1 xyz (ii) Compute the first few terms of the quantum period and the Picard Fuchs differential operator. (iii) Show that f is mirror to the 4-dimensional toric hypersurface X of type (, 1) in the toric variety F with weight data: (what IS this variety?) Appendix Table : Complete intersection descriptions of the representatives of Table 1 for the 6 mutation-equivalence classes with singularity content (n, {m 1 (1, 1)}). # Weights and line bundles

15 none

16

17 Table 1: Representatives for the 6 mutation-equivalence classes of Fano polygons P N Q with singularity content (n, {m 1 (1, 1)}). The degrees K X = 1 n 5m of the corresponding toric varieties are also given. See also Figure 4. # V(P ) n m K X 1 (7, 5), (, 5), (, 5) (, ), (, ), (, ), (, ) 8 (, 1), (, ), ( 1, ), (, 1), (, ), ( 1, ) (, ), ( 1, ), (, 1), (, ) (, 1), (1, ), ( 1, ), (, 1), (, 1), ( 1, ), (1, ), (, 1) (, ), ( 1, ), (, 1), (, 1), ( 1, ) (, 1), (1, ), ( 1, ), (, 1), (, 1), ( 1, ), (1, 1) (, 1), (1, ), ( 1, ), (, 1), (, 1), ( 1, ) 5 9 (1, 1), ( 1, ), (, 1), ( 1, 1), (1, ), (, 1) (1, 1), ( 1, ), ( 1, ), (1, ) (1, 1), ( 1, ), (, 1), ( 1, 1), (, 1) (, 1), (, 1), (0, 1) (1, 1), ( 1, ), ( 1, 1), (, 1) (1, 1), ( 1, ), (, 1), ( 1, 1), (1, 1) 4 15 (1, 1), ( 1, ), ( 1, 1), (1, 1) (1, 1), ( 1, ), ( 1, 0), (0, 1), (, 1) (1, 0), (1, 1), ( 1, ), (, 1), ( 1, 1), (0, 1) 4 18 (1, 0), (0, 1), ( 1, 1), ( 1, ) (1, 1), ( 1, ), ( 1, 1), (0, 1), (, 1) (1, 1), ( 1, ), (, 1), ( 1, 1), (0, 1) 5 1 (1, 1), ( 1, ), ( 1, ) (1, 1), ( 1, ), ( 1, 1), (0, 1) (1, 1), ( 1, ), (0, 1), (, 1) 17 4 (0, 1), ( 1, ), (, 1), ( 1, 0), (1, 1) (0, 1), ( 1, ), (, 1), (1, 1) 1 6 ( 1, ), (, 1), (1, 1)

18 Figure 4: Representatives for the 6 mutation-equivalence classes of Fano polygons P N Q with singularity content (n, {m 1 (1, 1)}). See also Table

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