The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6

The Pennsylvania State University The Graduate School Eberly College of Science AN ASYMPTOTIC MUKAI MODEL OF M 6 A Dissertation in Mathematics by Evgeny Mayanskiy c 2013 Evgeny Mayanskiy Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2013

The dissertation of Evgeny Mayanskiy was reviewed and approved by the following: Yuri Zarhin Professor of Mathematics Dissertation Advisor, Chair of Committee Robert Vaughan Professor of Mathematics Dale Brownawell Distinguished Professor of Mathematics Karl Schwede Assistant Professor of Mathematics Murat Gunaydin Professor of Physics Svetlana Katok Professor of Mathematics Chair of Graduate Program Signatures are on file in the Graduate School.

Abstract We study the Mukai construction of a general curve of genus 6 as a complete intersection of the Grassmannian of lines in P 4 with a codimension 5 quadric in the Plücker space. We formulate the relevant GIT problem in general and then solve it for the large values of the GIT parameter. This allows us to conclude that asymptotically Mukai compact model of M 6 parametrizes double anticanonical curves on the smooth del Pezzo surface of degree 5. As a byproduct of our study we obtain an explicit geometric interpretation of Ozeki classification of orbits of a certain prehomogeneous space. This complements earlier results of J.A. Todd [17]. iii

Table of Contents List of Figures Acknowledgments vi vii Chapter 1 Introduction 1 1.1 Preface................................... 1 1.2 Statement of the problem........................ 2 Chapter 2 The parameter space 3 Chapter 3 Hilbert-Mumford criterion 10 Chapter 4 The asymptotic case 21 Appendix A Classification of codimension 4 linear sections of the Grassmannian of lines in P 4 32 Appendix B Grassmannian degenerations of the del Pezzo quintic threefold 42 Appendix C Excel macros which computes extremal destabilizing 1-PS 47 iv

Appendix D Excel macros which identifies minimal sets of vanishing Plücker coordinates for unstable points 55 Appendix E MATLAB code which checks if a given matrix lies in the general orbit of GL(4) SL(5) 60 Bibliography 63 v

List of Figures 2.1 Two families................................ 4 A.1 Codimension 4 sections of G(2, 5) P 9, orbits 1-5............ 35 A.2 Codimension 4 sections of G(2, 5) P 9, orbits 6-11........... 36 A.3 Codimension 4 sections of G(2, 5) P 9, orbits 12-17.......... 37 A.4 Codimension 4 sections of G(2, 5) P 9, orbits 18-22.......... 38 A.5 Codimension 4 sections of G(2, 5) P 9, orbits 23-28.......... 39 A.6 Codimension 4 sections of G(2, 5) P 9, orbits 29-35.......... 40 A.7 Codimension 4 sections of G(2, 5) P 9, orbits 36-38.......... 41 B.1 Codimension 3 sections of G(2, 5) P 9, orbits 1-5............ 43 B.2 Codimension 3 sections of G(2, 5) P 9, orbits 6-10........... 44 B.3 Codimension 3 sections of G(2, 5) P 9, orbits 11-14.......... 45 B.4 Codimension 3 sections of G(2, 5) P 9, orbits 15-21.......... 46 vi

Acknowledgments Our work was made possible because of the Teaching Assistantship support from the Mathematical Department of the Pennsylvania State University. This work is a part of a larger project (joint with Damiano Fulghesu), where we study Mukai models of moduli spaces of curves of low genera and their relations with the Hassett-Keel program. We started working on it during the AIM Workshop Log minimal model program for moduli spaces in December 2012. We thank organizers of the workshop and staff of the American Institute of Mathematics for providing excellent working conditions and especially Brendan Hassett, who pointed out this problem during the workshop. Two months after we obtained the results presented in this thesis an entirely independent work by Fabian Müller appeared 1 which identified our asymptotic GIT quotient with the final log canonical model of M 6. 1 Fabian Müller, The final log canonical model of M 6, arxiv: 1303.6843 (2013). vii

Chapter 1 Introduction 1.1 Preface This thesis contains the beginnings of the study of Mukai models of the moduli space of curves of genus 6. Mukai-like constructions of curves, surfaces, threefolds... as complete intersections and sections of Grassmannians suggest that Geometric Invariant Theory can be used in order to construct quasiprojective moduli spaces of such varieties as well as their natural compactifications. Nevertheless, it appears that this application of Mumford s GIT did not attract much attention and was not explored much. We would like to point out [2], [3], [4], [7] among other works in this direction. In this thesis we study one such GIT construction of a moduli space of curves of genus 6. We hope that the methods we use and observations we make will be useful for other analogous GIT constructions involving sections of Grassmannians. As a byproduct of our study we obtain an explicit geometric classification of codimension 4 linear sections of the Grassmannian of lines in P 4. Linear sections of Grassmannians have a very beautiful geometry (see [11], [16], [15], [8], [6], [1] among many others). Our result is an elementary consequence of the Ozeki classification of orbits of a certain prehomogeneous space [14]. The problem of classifying linear sections (in particular, linear sections of codimensions 3 and 4) of the Grassmannian of lines in P 4 was addressed earlier by J.A. Todd [17]. We believe that our results complement his study.

2 1.2 Statement of the problem Let us recall the following result of Mukai (see section 5 of [12]). Theorem. (Mukai, [12]) Let C be a general curve of genus 6. Then: (1) there exists a unique stable rank 2 vector bundle F max on C such that det(f max ) = ω C is the canonical bundle and h 0 (C, F max ) = 5 is maximal possible, (2) F max is generated by global sections, (3) the morphism C G(2, H 0 (C, F max )) defined by the global sections of F max is a closed embedding which represents C as a complete intersection of G(2, H 0 (C, F max )) with a codimension 5 quadric in P( 2 H 0 (C, F max )) (where the Grassmannian is embedded via the Plücker embedding). Vice versa, it is easy to see that a general such complete intersection is a canonical curve of genus 6. This suggests the following natural question. Problem. Describe the quotient of a compactification of a parameter space of complete intersections G(2, V ) (codim 5 quadric in P( 2 V )) by the automorphism group Aut(V ) and represent this quotient as a compact model of M 6.

Chapter 2 The parameter space In this chapter we define a compactification X of a parameter space of complete intersections as above. Let k = C and V = k 5 with the standard action of G = SL(5). Let G(2, V ) P( 2 V ) be the Plücker embedding of the Grassmannian of affine 2-planes in V. We recall that the ideal I P of G(2, V ) as a subvariety of P( 2 V ) can be generated by quadratic polynomials p 1 = Z 0 Z 7 Z 1 Z 5 + Z 2 Z 4, p 2 = Z 0 Z 8 Z 1 Z 6 + Z 3 Z 4, p 3 = Z 0 Z 9 Z 2 Z 6 + Z 3 Z 5, p 4 = Z 1 Z 9 Z 2 Z 8 + Z 3 Z 7, p 5 = Z 4 Z 9 Z 5 Z 8 + Z 6 Z 7, where Z 0, Z 1, Z 2, Z 3, Z 4, Z 5, Z 6, Z 7, Z 8, Z 9 are the homogeneous coordinates on P( 2 V ) = P 9. If X 1, X 2, X 3, X 4, X 5 are coordinates of V, then one can take Z 0 = X 1 X 2, Z 1 = X 1 X 3, Z 2 = X 1 X 4, Z 3 = X 1 X 5, Z 4 = X 2 X 3, Z 5 = X 2 X 4, Z 6 = X 2 X 5, Z 7 = X 3 X 4, Z 8 = X 3 X 5, Z 9 = X 4 X 5. Consider the Grassmannian G(4, H 0 (G(2, V ), O G(2,V ) (1))) = G(4, 2 V ) of codi-

4 Figure 2.1. Two families. mension 4 linear subspaces in P( 2 V ). We will sometimes restrict ourselves to open subsets U = Spec(k[a ij ]) G(4, 2 V ) corresponding to linear subspaces of P( 2 V ) given by 4 equations Z ik j i 1,i 2,i 3,i 4 a kj Z j, 1 k 4. Let us denote Π k = V + (Z ik j i 1,i 2,i 3,i 4 a kj Z j ) U P( 2 V ), 1 k 4. Let S be the tautological rank 6 vector bundle over G(4, 2 V ). Then P(S) G(4, 2 V ) P( 2 V ) is the universal family over G(4, 2 V ) and P(S) U = Π 1 Π 2 Π 3 Π 4 U P( 2 V ). Consider the trivial family G(4, 2 V ) G(2, V ) G(4, 2 V ) P( 2 V ). Lemma 1. The intersection P(S) (G(4, 2 V ) G(2, V )) is a smooth connected subvariety of G(4, 2 V ) P( 2 V ). Proof: It is sufficient to work over U G(4, 2 V ) as above. We may assume that Z 9 = 1 by restricting to D + (Z 9 ) U P( 2 V ). Without loss of generality we may assume that 9 i 1, i 2, i 3, i 4. Then P(S) U D+ (Z 9 ) is given inside of D + (Z 9 ) = Spec(k[a ij ][Z 0, Z 1, Z 2, Z 3, Z 4, Z 5, Z 6, Z 7, Z 8 ])

5 by equations a k9 + a kj Z j Z ik, 1 k 4, j i 1,i 2,i 3,i 4,9 while U G(2, V ) D + (Z 9 ) is given by equations Z 0 = (Z 2 Z 6 Z 3 Z 5 ), Z 1 = (Z 2 Z 8 Z 3 Z 7 ), Z 4 = (Z 5 Z 8 Z 6 Z 7 ). QED Let us denote by i: P(S) G(4, 2 V ) P( 2 V ) the closed embedding of the universal family π : P(S) G(4, 2 V ) over the Grassmannian. Lemma 2. The morphism of coherent sheaves i I P i O G(4, 2 V ) P( 2 V ) = O P(S) is injective. Proof: We use notation from the proof of Lemma 1. As in Lemma 1, we work in D + (Z 9 ) over U G(4, 2 V ). Let T = Π 1... Π i D + (Z 9 ) U P 9 and H = T Π i+1, 0 i 3. Then H T is a Cartier divisor and the corresponding invertible sheaf is O T (1). We will prove Lemma 2 by induction on i. If we multiply the short exact sequence 0 O T ( 1) O T O T H 0 by OT I P, then we get a short exact sequence 0 I P ( 1) T I P T I P T H 0. The injectivity of the morphism I P ( 1) T I P T follows from the induction assumption that I P T O T is injective. Then a diagram chase shows that I P T H O T H is injective as well. QED

6 Lemma 3. R 1 π (i I P (2)) = 0. Proof: As in Lemma 1, we work over U G(4, 2 V ). Then it is sufficient to show that H 1 (P(S) U, i I P (2)) = 0. We use notation from the proof of Lemma 2 and do induction on i {0, 1, 2, 3}. For any d Z we have the short exact sequence of coherent sheaves on T 0 I P T (d 1) I P T (d) I P T H (d) 0. Hence vanishing of H i (T H, I P T H (3 i)) follows from vanishing of H i (T, I P T (3 i)) and H i+1 (T, I P T (3 i 1)). By the cohomological flat base change theorem H i (U P 9, i I P (d)) = k[a ij ] k H i (P 9, I P (d)). So, it is enough to check that for the Plücker ideal I P as a coherent sheaf on P 9 we have H i (P 9, I P (3 i)) = 0 for 1 i 5. Let P = G(2, V ) P( 2 V ) = P 9 be the Plücker embedding. The short exact sequence 0 I P (d) O P 9(d) O P (d) 0 and projective normality of the Plücker embedding imply that H 1 (P 9, I P (d)) = 0 and H i (P 9, I P (d)) = H i 1 (P, O P (d)), 2 i 8 for any d Z. Finally, H i (P, O P (2 i)) = H i (P, ω P (7 i)) = 0, 1 i 4 by the Kodaira Vanishing theorem. QED

7 Definition 1. E = π (O(2)/i I P (2)). Corollary 1. There exists a surjective morphism of coherent sheaves π (O(2)) = Sym 2 (S ) E with kernel π (i I P (2)). Let us describe this kernel more explicitely over an open subset U G(4, 2 V ) as above. Lemma 4. Over U G(4, 2 V ) one can describe π (i I P (2)) U as a subsheaf of π (O(2)) U = Sym 2 (S ) U = O U Z i Z j 0 i j 9, i,j i 1,i 2,i 3,i 4 as follows: π (i I P (2)) U = 5 O U p i π (O(2)) U. i=1 Proof: We will be working in the graded ring B = k[a ij ][Z 0,..., ˆ Zi1,..., ˆ Zi2,..., ˆ Zi3,..., ˆ Zi4,..., Z 9 ]. Note that P(S) U = P roj(b) U P 9. Let ξ H 0 (P(S) U, i I P (2)). Then ξ D+ (Z i ) = 5 j=1 ω ij (Z i ) N p j for some ω ij B N homogeneous elements of B of degree N for any i. Comparing ξ D+ (Z i ) and ξ D + (Z i we get ) 5 ω ij p j = (Z i ) N γ i j=1

8 in B for some γ i B 2. Since i I P B is a prime ideal by Lemma 1, we conclude that γ i i I P. Hence ξ D+ (Z i ) = 5 j=1 γ j p j for some γ j k[a ij ] for any i. QED Remark 1. It follows from [14] that E is a locally free sheaf of rank 16 over the open subset of G(4, 2 V ), which is the complement to the union of the following G-orbits (we use the numeration of G-orbits in G(4, 2 V ) from [14]): orbits 29, 32, 36, over which the dimension of the fibers of E is 17, orbits 33, 35, 37, over which the dimension of the fibers of E is 18, orbit 38, over which the dimension of the fibers of E is 20. As a compactification of a parameter space of complete intersections G(2, V ) ( codim 5 quadric in P( 2 V )) we take the following space X. Definition 2. X = P(E). Let us denote by p: X = P(E) G(4, 2 V ) the canonical projection. Note that G acts on X and p is G-equivariant. Corollary 2. The action of G on X extends to an action of G on the tautological line bundle O(1) on X. Proof: The Plücker ideal I P O P( 2 V ) is G-invariant. Hence G acts on E. Hence it acts on the space of lines in its fibers which is the tautological line bundle O(1). QED We will study the following linearizations of G = SL(5)-action on X.

9 Definition 3. L a,b = O P(E) (a) O p O G(4, 2 V )(b), a, b Z, a, b > 0. We will also use notation L a,b = L t, where t = b/a. In this thesis we will focus on the asymptotic case, when t.

Chapter 3 Hilbert-Mumford criterion In this chapter we give a solution to the Hilbert-Mumford numertical criterion of stability for the GIT problem described in Chapter 2. Let λ: G m G = SL(5) be a 1-PS. We can assume that we have chosen coordinates X 1, X 2, X 3, X 4, X 5 in such a way that λ diagonalizes with weights w 1, w 2, w 3, w 4, w 5 such that w 1 w 2 w 3 w 4 w 5 and w 1 + w 2 + w 3 + w 4 + w 5 = 0. Let us denote by τ 0 = w 1 + w 2, τ 1 = w 1 + w 3, τ 2 = w 1 + w 4, τ 3 = w 1 + w 5, τ 4 = w 2 + w 3, τ 5 = w 2 + w 4, τ 6 = w 2 + w 5, τ 7 = w 3 + w 4, τ 8 = w 3 + w 5, τ 9 = w 4 + w 5. the corresponding weights of the coordinates Z i on P( 2 V ) = P 9. It follows from Lemma 4 that a point ξ X is a quadric q in the 6-dimensional projective space π = p(ξ) G(4, 2 V ) taken modulo Plücker quadrics p 1 = Z 0 Z 7 Z 1 Z 5 + Z 2 Z 4, p 2 = Z 0 Z 8 Z 1 Z 6 + Z 3 Z 4, p 3 = Z 0 Z 9 Z 2 Z 6 + Z 3 Z 5, p 4 = Z 1 Z 9 Z 2 Z 8 + Z 3 Z 7, p 5 = Z 4 Z 9 Z 5 Z 8 + Z 6 Z 7.

11 We will write ξ = (π, q). Then a standard computation (see [13] or [5]) shows that the Hilbert-Mumford weight function of ξ can be expressed as follows: µ(ξ, λ) = a (τ i + τ j ) + b (τ i1 + τ i2 + τ i3 + τ i4 ), where τ i1 +τ i2 +τ i3 +τ i4 is the maximal λ-weight at π of the action of G on G(4, 2 V ) and τ i + τ j is the maximal λ-weight of the quadratic polynomial q (we take its representative modulo p 1, p 2, p 3, p 4, p 5 with the minimal such weight). ([13]): Then the Hilbert-Mumford numerical criterion of stability can be stated as follows ξ is semistable if and only if µ(ξ, λ) 0 for any λ, (3.1) ξ is properly stable if and only if µ(ξ, λ) > 0 for any λ, (3.2) ξ is unstable if and only if µ(ξ, λ) < 0 for some λ. (3.3) Lemma 5. Given a maximal torus in G, there exists a finite set of 1-PS diagonalized by this maximal torus for which it is sufficient to check inequalities (3.1) (3.3). Proof: Let us fix ξ = (π, q) X. Let us denote by τ the ordering induced on τ i by a given 1-PS λ with weights w 1, w 2, w 3, w 4, w 5. Let τ i0 τ i1 τ i2 τ i3 τ i4 τ i5 τ i6 τ i7 τ i8 τ i9. Let us denote by τ 2 the ordering induced on τ I = τ i + τ j, i j, I = {i, j}. We will choose indices in such a way that τ Ii+1 τ Ii τ I0 for any i 0. The choice of coordinates X i in V determines the flag 0 = E 0 E 1 E 2 E 3 E 4 E 5 = V, where E i V is the linear subspace cut out by equations X i+1 =... = X 5 = 0. The ordering τ determines a flag in P( 2 V ). The ordering τ 2 determines a flag in the vector space H 0 (P π, O(2)) of quadrics in the 5-dimensional linear subspace P π P( 2 V ) = P 9 cut out by the equations π G(4, 2 V ). Let us notice (see [13] (section 4 in chapter 4) or [5] (chapter 11)) that one can

12 write µ(ξ, λ) in the form µ(ξ, λ) = a ( i 0 f i (τ Ii+1 τ Ii ) + f 0 τ I0 ) + b ( j 0 d j (τ ij τ ij+1 ) + d 9 τ 9 ), where f i 0 are the dimensions of intersections of the line determined by the quadric q with the vector spaces of the flag in H 0 (P π, O(2)) defined above and d j 0 are the dimensions of intersections of the 4-dimensional space determined by π with the vector spaces of the flag in P( 2 V ) defined above. Orderings τ and τ 2 determine a convext polytope in the space of w j. Since τ i and τ I are linear in w j we can express them as nonnegative linear combinations of their values at w j corresponding to vertices of this convext polytope. Hence the same holds for µ(ξ, λ): it is equal to a linear combination with nonnegative coefficients of µ(ξ, λ k ), where λ k are 1-PS with the same diagonalizing maximal torus in G, but whose weights correspond to the (finitely many) vertices of the convex polytope above. This proves the claim. QED Let us compute these extremal destabilizing 1-PS explicitly. It is sufficient to find all the vertices of convex polytopes whose sides are given by linear equations (in coordinates w i ) of the form τ i = τ j and τ i + τ j = τ k + τ l for various indices i, j, k, l. Let us switch to more convenient coordinates first: c 1 = w 1 w 2 5 Then c i 0 for any i and, c 2 = w 2 w 3 5, c 3 = w 3 w 4 5, c 4 = w 4 w 5. 5 w 1 = 4c 1 + 3c 2 + 2c 3 + c 4, w 2 = c 1 + 3c 2 + 2c 3 + c 4, w 3 = c 1 2c 2 + 2c 3 + c 4, w 4 = c 1 2c 2 3c 3 + c 4, w 5 = c 1 2c 2 3c 3 4c 4, τ 0 = 3c 1 + 6c 2 + 4c 3 + 2c 4, τ 1 = 3c 1 + c 2 + 4c 3 + 2c 4, τ 2 = 3c 1 + c 2 c 3 + 2c 4, τ 3 = 3c 1 + c 2 c 3 3c 4, τ 4 = 2c 1 + c 2 + 4c 3 + 2c 4, τ 5 = 2c 1 + c 2 c 3 + 2c 4, τ 6 = 2c 1 + c 2 c 3 3c 4, τ 7 = 2c 1 4c 2 c 3 + 2c 4, τ 8 = 2c 1 4c 2 c 3 3c 4, τ 9 = 2c 1 4c 2 6c 3 3c 4.

13 Note that we can talk either about convex polytopes in the projective space of c i s or about convex cones in the affine space of c i s. All possible equations of the form τ i = τ j and τ i + τ j = τ k + τ l can be written in terms of c i as follows (we list the 4-tuples of coefficients (u 1, u 2, u 3, u 4 ) of equations u 1 c 1 + u 2 c 2 + u 3 c 3 + u 4 c 4 = 0): (1, 1, 0, 0) (1, 0, 1, 0) (1, 0, 0, 1) (0, 1, 1, 0) (0, 1, 0, 1) (0, 0, 1, 1) (2, 0, 0, 1) (1, 0, 0, 2) (2, 0, 1, 0) (1, 0, 2, 0) (0, 2, 0, 1) (0, 1, 0, 2) (0, 1, 1, 1) (0, 1, 1, 1) (1, 1, 2, 0) (2, 1, 1, 0) (1, 1, 1, 0) (1, 0, 2, 2) (2, 0, 1, 1) (1, 2, 0, 2) (1, 1, 0, 2) (1, 1, 0, 2) (2, 1, 0, 2) (1, 0, 1, 2) (1, 0, 1, 1) (1, 0, 1, 1) (1, 2, 0, 1) (1, 1, 0, 1) (1, 0, 1, 1) (1, 0, 2, 1) (1, 1, 0, 1) (1, 1, 0, 1) (2, 1, 0, 1) (2, 0, 1, 2) (2, 0, 1, 1) (2, 0, 1, 1) (2, 0, 2, 1) (1, 1, 0, 2) (2, 2, 0, 1) (0, 1, 1, 1) (0, 1, 1, 2) (0, 2, 1, 1) (1, 1, 1, 0) (1, 1, 1, 0) (1, 1, 1, 2) (2, 1, 1, 1) (1, 1, 1, 1) (2, 1, 1, 2) (1, 1, 1, 1) (1, 1, 3, 1) (2, 3, 1, 1) (1, 1, 3, 2) (1, 2, 1, 1) (1, 1, 2, 1) (1, 1, 1, 1) (1, 3, 1, 1) (1, 1, 1, 1) (1, 1, 2, 2) (2, 2, 1, 1) (2, 1, 1, 1) (1, 1, 1, 2) (1, 1, 1, 1) (1, 1, 1, 1) (1, 1, 1, 2) (2, 1, 1, 1) (1, 1, 1, 1) (1, 0, 0, 0) (0, 1, 0, 0) (0, 0, 1, 0) (0, 0, 0, 1). Now one has to go over all triples of these equations and whenever their common solution is a line in the first hyperoctant of the space of c i, this solution will be a vertex of a convex polytope (or an extremal ray of a convex cone) as above. We used an Excel macros (see Appendix C) to do this computation and obtained the following list of vertices (we list the representative nonzero 4-tuples (c 1, c 2, c 3, c 4 )): (1, 1, 1, 1) (1, 1, 1, 2) (2, 2, 2, 1) (0, 0, 0, 1) (2, 2, 2, 3) (1, 1, 1, 3) (1, 1, 1, 4) (1, 1, 1, 5) (1, 1, 1, 6) (1, 1, 2, 1) (2, 2, 1, 2) (1, 1, 0, 1) (1, 1, 3, 1) (1, 1, 4, 1) (1, 1, 2, 2) (2, 2, 1, 1) (0, 0, 1, 1) (1, 1, 3, 3) (4, 4, 1, 1) (2, 2, 3, 3) (3, 3, 1, 1)

14 (3, 3, 2, 2) (1, 1, 4, 4) (2, 2, 1, 4) (1, 1, 0, 2) (1, 1, 3, 2) (1, 1, 4, 2) (1, 1, 5, 2) (2, 2, 4, 1) (2, 2, 0, 1) (2, 2, 6, 1) (2, 2, 3, 1) (4, 4, 1, 2) (2, 2, 5, 1) (4, 4, 3, 2) (2, 2, 7, 1) (1, 1, 2, 3) (2, 2, 4, 3) (1, 1, 2, 4) (1, 1, 2, 6) (1, 1, 2, 7) (1, 1, 2, 5) (2, 2, 1, 3) (4, 4, 2, 1) (2, 2, 1, 6) (4, 4, 2, 3) (2, 2, 1, 5) (2, 2, 1, 8) (4, 4, 2, 5) (2, 2, 1, 9) (2, 2, 1, 11) (2, 2, 1, 7) (1, 1, 3, 4) (3, 3, 1, 4) (2, 2, 0, 3) (1, 1, 0, 3) (1, 1, 0, 4) (1, 1, 0, 5) (2, 2, 6, 3) (1, 1, 3, 5) (1, 1, 3, 7) (1, 1, 3, 8) (1, 1, 3, 6) (2, 2, 5, 3) (2, 2, 7, 3) (4, 4, 1, 6) (2, 2, 9, 3) (1, 1, 4, 3) (3, 3, 4, 1) (1, 1, 5, 4) (3, 3, 5, 2) (1, 1, 5, 3) (1, 1, 6, 3) (1, 1, 6, 4) (3, 3, 7, 1) (1, 1, 7, 4) (2, 1, 2, 2) (1, 2, 1, 1) (1, 0, 1, 1) (1, 3, 1, 1) (1, 4, 1, 1) (1, 2, 1, 2) (2, 1, 2, 1) (0, 1, 0, 1) (3, 1, 3, 1) (1, 3, 1, 3) (1, 4, 1, 2) (1, 3, 1, 2) (2, 3, 2, 4) (1, 5, 1, 2) (2, 1, 2, 4) (1, 0, 1, 2) (1, 6, 1, 2) (4, 1, 4, 2) (2, 3, 2, 1) (2, 4, 2, 1) (2, 0, 2, 1) (2, 6, 2, 1) (2, 5, 2, 1) (0, 1, 0, 2) (0, 2, 0, 1) (0, 1, 0, 3) (1, 2, 1, 4) (5, 1, 5, 2) (3, 2, 3, 4) (3, 1, 3, 2) (2, 3, 2, 6) (1, 3, 1, 6) (4, 2, 4, 1) (3, 2, 3, 1) (1, 6, 1, 3) (3, 4, 3, 2) (1, 2, 1, 3) (2, 1, 2, 3) (1, 4, 1, 3) (2, 4, 2, 5) (1, 2, 1, 5) (2, 4, 2, 3) (1, 2, 1, 6) (1, 2, 1, 8) (1, 2, 1, 9) (1, 2, 1, 7) (4, 1, 4, 3) (2, 5, 2, 6) (4, 3, 4, 5) (1, 7, 1, 3) (3, 5, 3, 1) (1, 6, 1, 4) (3, 1, 3, 5) (1, 5, 1, 3) (2, 1, 2, 6) (1, 0, 1, 3) (1, 8, 1, 3) (3, 1, 3, 9) (2, 1, 2, 7) (1, 2, 2, 1) (2, 1, 1, 2) (3, 1, 1, 3) (1, 3, 3, 1) (1, 0, 0, 1) (2, 1, 4, 2) (1, 3, 2, 1) (1, 0, 2, 1) (1, 4, 2, 1) (1, 5, 2, 1) (2, 4, 1, 2) (2, 3, 1, 2) (2, 0, 1, 2) (2, 6, 1, 2) (2, 5, 1, 2) (4, 1, 2, 4) (2, 7, 1, 2) (4, 2, 1, 4) (2, 1, 5, 2) (2, 1, 0, 2) (2, 1, 6, 2) (2, 1, 3, 2) (2, 1, 7, 2) (1, 2, 4, 1) (1, 2, 0, 1) (1, 2, 3, 1) (1, 2, 5, 1) (1, 4, 3, 1) (1, 3, 5, 1) (1, 3, 0, 1) (1, 3, 4, 1) (1, 3, 6, 1) (1, 0, 3, 1) (1, 5, 3, 1) (1, 6, 3, 1) (1, 2, 2, 2) (2, 1, 1, 1) (3, 1, 1, 1) (0, 1, 1, 1) (4, 1, 1, 1)

15 (3, 2, 2, 2) (5, 1, 1, 1) (6, 1, 1, 1) (2, 1, 1, 4) (1, 4, 4, 2) (3, 1, 1, 6) (1, 0, 0, 2) (2, 3, 3, 4) (1, 5, 5, 2) (1, 3, 3, 2) (3, 2, 2, 6) (4, 1, 1, 8) (2, 4, 4, 1) (4, 1, 1, 2) (6, 2, 2, 3) (2, 0, 0, 1) (2, 3, 3, 1) (2, 5, 5, 1) (4, 3, 3, 2) (6, 1, 1, 3) (8, 1, 1, 4) (1, 2, 2, 4) (2, 4, 4, 5) (1, 2, 2, 3) (1, 2, 2, 5) (2, 4, 4, 3) (1, 2, 2, 6) (1, 2, 2, 9) (1, 2, 2, 10) (1, 2, 2, 7) (1, 2, 2, 8) (4, 2, 2, 1) (2, 1, 1, 3) (4, 2, 2, 5) (2, 1, 1, 5) (4, 2, 2, 3) (2, 1, 1, 6) (2, 1, 1, 8) (2, 1, 1, 7) (3, 1, 1, 2) (0, 1, 1, 2) (6, 1, 1, 2) (3, 2, 2, 4) (5, 1, 1, 2) (5, 2, 2, 4) (8, 1, 1, 2) (7, 1, 1, 2) (6, 2, 2, 1) (0, 2, 2, 1) (3, 4, 4, 2) (3, 2, 2, 1) (5, 2, 2, 1) (5, 4, 4, 2) (9, 2, 2, 1) (10, 2, 2, 1) (7, 2, 2, 1) (8, 2, 2, 1) (3, 1, 1, 5) (6, 2, 2, 5) (6, 2, 2, 7) (3, 1, 1, 4) (3, 1, 1, 7) (3, 1, 1, 8) (3, 1, 1, 10) (3, 1, 1, 9) (0, 1, 1, 3) (0, 1, 1, 4) (8, 1, 1, 3) (4, 3, 3, 5) (3, 5, 5, 1) (5, 1, 1, 9) (4, 1, 1, 7) (4, 1, 1, 3) (6, 1, 1, 4) (10, 1, 1, 6) (8, 1, 1, 5) (4, 6, 6, 1) (7, 1, 1, 3) (5, 3, 3, 1) (1, 4, 4, 3) (1, 6, 6, 4) (5, 2, 2, 6) (3, 4, 4, 5) (3, 4, 4, 1) (4, 1, 1, 6) (5, 1, 1, 3) (1, 3, 3, 5) (7, 2, 2, 6) (5, 4, 4, 3) (1, 5, 5, 3) (5, 3, 3, 4) (6, 1, 1, 10) (5, 1, 1, 8) (9, 1, 1, 5) (7, 1, 1, 4) (10, 1, 1, 3) (9, 1, 1, 3) (2, 4, 1, 4) (1, 2, 0, 2) (1, 2, 4, 2) (1, 2, 3, 2) (1, 2, 6, 2) (1, 2, 5, 2) (2, 1, 4, 1) (2, 1, 0, 1) (4, 2, 1, 2) (2, 1, 5, 1) (2, 1, 3, 1) (4, 2, 3, 2) (2, 1, 6, 1) (1, 3, 2, 3) (3, 1, 6, 1) (1, 4, 2, 4) (2, 3, 4, 3) (2, 3, 1, 3) (6, 2, 3, 2) (4, 1, 2, 1) (4, 3, 2, 3) (2, 5, 1, 5) (6, 1, 3, 1) (3, 1, 0, 1) (5, 1, 2, 1) (5, 3, 1, 3) (3, 1, 7, 1) (1, 3, 5, 3) (3, 4, 1, 4) (6, 2, 1, 2) (6, 1, 2, 1) (3, 1, 5, 1) (3, 2, 4, 2) (0, 1, 2, 1) (3, 1, 4, 1) (3, 1, 2, 1) (6, 2, 5, 2) (3, 1, 8, 1) (9, 3, 1, 3) (3, 2, 1, 2) (4, 1, 6, 1) (5, 2, 4, 2) (5, 4, 3, 4) (8, 1, 2, 1) (9, 1, 2, 1) (7, 1, 2, 1) (7, 2, 1, 2) (1, 4, 2, 2) (1, 0, 2, 2) (1, 3, 2, 2) (2, 3, 4, 4) (1, 5, 2, 2) (2, 1, 4, 4) (1, 6, 2, 2) (1, 7, 2, 2) (4, 1, 2, 2) (2, 0, 1, 1) (2, 3, 1, 1) (2, 4, 1, 1) (2, 5, 1, 1) (5, 1, 2, 2) (1, 2, 4, 4) (8, 1, 2, 2) (3, 2, 4, 4)

16 (6, 1, 2, 2) (0, 1, 2, 2) (3, 1, 2, 2) (5, 2, 4, 4) (9, 1, 2, 2) (11, 1, 2, 2) (7, 1, 2, 2) (4, 2, 1, 1) (0, 2, 1, 1) (3, 2, 1, 1) (3, 4, 2, 2) (6, 2, 1, 1) (7, 2, 1, 1) (5, 2, 1, 1) (4, 0, 1, 1) (3, 0, 1, 1) (3, 0, 2, 2) (5, 0, 1, 1) (5, 3, 1, 1) (4, 5, 1, 1) (1, 4, 3, 3) (3, 4, 1, 1) (3, 5, 2, 2) (2, 5, 3, 3) (4, 6, 1, 1) (4, 3, 1, 1) (4, 7, 1, 1) (4, 1, 3, 3) (6, 1, 4, 4) (3, 5, 1, 1) (3, 7, 2, 2) (1, 7, 3, 3) (3, 6, 1, 1) (0, 3, 1, 1) (3, 6, 2, 2) (3, 9, 2, 2) (7, 3, 1, 1) (8, 3, 1, 1) (6, 3, 1, 1) (2, 8, 1, 4) (2, 3, 1, 4) (2, 5, 1, 4) (2, 0, 1, 4) (2, 10, 1, 4) (2, 6, 1, 4) (2, 9, 1, 4) (4, 3, 2, 8) (2, 11, 1, 4) (6, 1, 3, 12) (4, 1, 2, 8) (2, 7, 1, 4) (1, 4, 6, 2) (1, 4, 3, 2) (1, 4, 0, 2) (1, 4, 5, 2) (1, 4, 8, 2) (1, 4, 7, 2) (2, 1, 3, 4) (1, 5, 3, 2) (1, 6, 4, 2) (4, 2, 1, 8) (5, 1, 2, 10) (2, 3, 7, 4) (1, 5, 7, 2) (2, 1, 5, 4) (1, 3, 5, 2) (1, 5, 4, 2) (2, 3, 0, 4) (1, 5, 0, 2) (2, 1, 0, 4) (1, 3, 0, 2) (3, 1, 0, 6) (2, 3, 6, 4) (2, 3, 8, 4) (2, 3, 5, 4) (2, 3, 11, 4) (2, 3, 9, 4) (1, 5, 6, 2) (1, 5, 9, 2) (1, 5, 8, 2) (2, 1, 6, 4) (1, 0, 3, 2) (1, 7, 3, 2) (1, 8, 3, 2) (1, 6, 3, 2) (2, 1, 8, 4) (2, 1, 9, 4) (2, 1, 7, 4) (1, 3, 4, 2) (1, 3, 7, 2) (1, 3, 6, 2) (1, 0, 4, 2) (1, 8, 4, 2) (1, 9, 4, 2) (1, 7, 4, 2) (3, 1, 4, 6) (4, 1, 8, 2) (2, 5, 4, 1) (2, 0, 4, 1) (2, 6, 4, 1) (2, 3, 4, 1) (2, 8, 4, 1) (2, 7, 4, 1) (8, 2, 3, 4) (4, 1, 9, 2) (4, 1, 0, 2) (4, 1, 6, 2) (4, 1, 10, 2) (4, 1, 3, 2) (4, 1, 5, 2) (4, 1, 11, 2) (12, 3, 1, 6) (8, 2, 1, 4) (4, 1, 7, 2) (6, 4, 1, 3) (2, 4, 3, 1) (4, 3, 1, 2) (2, 6, 5, 1) (4, 5, 3, 2) (2, 4, 8, 1) (2, 3, 7, 1) (2, 5, 3, 1) (4, 5, 1, 2) (2, 7, 5, 1) (4, 7, 3, 2) (2, 4, 0, 1) (2, 3, 0, 1) (2, 0, 3, 1) (2, 7, 3, 1) (2, 6, 3, 1) (4, 0, 1, 2) (2, 0, 5, 1) (4, 0, 3, 2) (6, 0, 1, 3) (4, 8, 1, 2) (2, 4, 5, 1) (4, 8, 3, 2) (2, 4, 6, 1) (2, 4, 9, 1) (2, 4, 7, 1) (4, 6, 1, 2) (8, 1, 2, 4) (4, 9, 1, 2) (4, 7, 1, 2) (2, 3, 5, 1) (4, 6, 3, 2) (2, 3, 8, 1) (2, 3, 6, 1) (2, 9, 5, 1) (2, 8, 5, 1) (4, 11, 3, 2) (4, 9, 3, 2) (10, 2, 1, 5) (1, 3, 2, 6)

17 (5, 1, 10, 2) (3, 2, 6, 4) (2, 3, 4, 6) (3, 1, 6, 2) (1, 4, 2, 8) (2, 6, 4, 3) (4, 2, 8, 1) (1, 6, 2, 3) (3, 2, 6, 1) (1, 8, 2, 4) (3, 4, 6, 2) (1, 3, 2, 5) (2, 1, 4, 5) (1, 6, 2, 4) (1, 5, 2, 3) (1, 0, 2, 3) (1, 0, 2, 4) (2, 6, 4, 7) (2, 6, 4, 5) (1, 3, 2, 7) (1, 3, 2, 4) (1, 3, 2, 8) (1, 3, 2, 12) (1, 3, 2, 13) (1, 3, 2, 9) (1, 3, 2, 10) (2, 5, 4, 6) (4, 1, 8, 3) (2, 7, 4, 8) (4, 3, 8, 5) (4, 5, 8, 7) (1, 7, 2, 3) (1, 9, 2, 4) (3, 5, 6, 1) (3, 7, 6, 2) (1, 4, 2, 3) (3, 4, 6, 5) (1, 8, 2, 5) (2, 1, 4, 6) (1, 8, 2, 3) (1, 9, 2, 3) (2, 3, 4, 8) (3, 1, 6, 5) (3, 2, 6, 7) (2, 1, 4, 3) (1, 5, 2, 4) (1, 7, 2, 4) (1, 10, 2, 4) (1, 11, 2, 4) (3, 1, 6, 12) (2, 1, 4, 8) (3, 5, 6, 4) (3, 1, 6, 8) (3, 8, 6, 1) (2, 1, 4, 9) (2, 3, 1, 6) (4, 3, 2, 6) (10, 2, 5, 4) (6, 4, 3, 8) (8, 1, 4, 2) (6, 2, 3, 4) (2, 4, 1, 8) (8, 3, 4, 6) (4, 5, 2, 10) (6, 1, 3, 2) (10, 1, 5, 2) (2, 5, 1, 10) (4, 6, 2, 3) (2, 6, 1, 3) (6, 4, 3, 2) (2, 10, 1, 5) (6, 2, 3, 1) (8, 2, 4, 1) (4, 1, 2, 3) (2, 4, 1, 5) (2, 4, 1, 3) (6, 4, 3, 1) (2, 6, 1, 5) (4, 3, 2, 1) (4, 5, 2, 3) (2, 0, 1, 3) (4, 0, 2, 1) (4, 0, 2, 3) (2, 0, 1, 5) (4, 6, 2, 1) (4, 6, 2, 7) (2, 3, 1, 8) (2, 3, 1, 5) (2, 3, 1, 7) (4, 6, 2, 5) (2, 3, 1, 10) (2, 3, 1, 12) (2, 3, 1, 14) (2, 3, 1, 9) (2, 3, 1, 11) (2, 8, 1, 3) (4, 1, 2, 6) (2, 5, 1, 3) (2, 7, 1, 3) (2, 9, 1, 3) (4, 10, 2, 3) (2, 12, 1, 5) (6, 8, 3, 1) (8, 6, 4, 1) (8, 2, 4, 3) (2, 8, 1, 6) (6, 2, 3, 7) (2, 10, 1, 7) (4, 5, 2, 1) (4, 8, 2, 1) (4, 7, 2, 1) (4, 3, 2, 10) (6, 1, 3, 8) (4, 1, 2, 5) (4, 7, 2, 3) (2, 7, 1, 5) (6, 5, 3, 1) (6, 1, 3, 5) (4, 1, 2, 9) (4, 8, 2, 3) (8, 1, 4, 6) (4, 12, 2, 3) (4, 9, 2, 3) (2, 8, 1, 5) (4, 1, 2, 10) (2, 11, 1, 5) (2, 13, 1, 5) (6, 2, 3, 15) (2, 9, 1, 5) (6, 4, 3, 5) (6, 7, 3, 2) (2, 4, 1, 9) (6, 5, 3, 4) (2, 9, 1, 6) (6, 4, 1, 8) (7, 1, 3, 2) (5, 3, 1, 6) (7, 3, 2, 6) (9, 1, 4, 2) (5, 1, 11, 2) (3, 2, 8, 4) (1, 3, 5, 6) (3, 1, 7, 2) (1, 4, 6, 8) (2, 3, 7, 6) (10, 2, 1, 4) (10, 1, 3, 2) (5, 1, 8, 2) (3, 1, 4, 2) (5, 2, 6, 4) (5, 3, 4, 6) (5, 1, 7, 2) (5, 1, 3, 2) (10, 2, 3, 4) (5, 1, 6, 2) (5, 1, 12, 2) (5, 1, 4, 2) (5, 1, 0, 2) (5, 1, 13, 2) (15, 3, 2, 6) (5, 1, 9, 2)

18 (3, 2, 1, 4) (3, 2, 7, 4) (3, 2, 10, 4) (3, 2, 0, 4) (3, 2, 5, 4) (3, 2, 12, 4) (3, 2, 9, 4) (9, 2, 1, 4) (1, 2, 3, 4) (3, 1, 5, 2) (1, 2, 5, 4) (6, 1, 8, 2) (0, 1, 3, 2) (5, 2, 1, 4) (6, 2, 1, 4) (8, 1, 3, 2) (8, 3, 1, 6) (3, 1, 8, 2) (3, 1, 0, 2) (3, 1, 9, 2) (1, 2, 6, 4) (1, 2, 0, 4) (1, 2, 8, 4) (1, 2, 7, 4) (7, 2, 6, 4) (7, 1, 10, 2) (1, 3, 4, 6) (3, 4, 2, 8) (1, 3, 8, 6) (12, 1, 3, 2) (14, 1, 3, 2) (9, 1, 3, 2) (11, 1, 3, 2) (4, 3, 5, 6) (6, 1, 9, 2) (6, 4, 1, 2) (8, 2, 3, 1) (8, 6, 1, 3) (4, 2, 10, 1) (3, 2, 8, 1) (1, 6, 8, 3) (3, 4, 1, 2) (5, 6, 1, 3) (1, 6, 5, 3) (5, 8, 3, 4) (4, 2, 7, 1) (3, 2, 5, 1) (4, 6, 5, 3) (0, 2, 3, 1) (4, 2, 5, 1) (4, 2, 3, 1) (8, 4, 3, 2) (4, 2, 6, 1) (4, 2, 9, 1) (8, 4, 7, 2) (4, 2, 0, 1) (4, 2, 11, 1) (12, 6, 1, 3) (8, 4, 1, 2) (5, 2, 3, 1) (5, 4, 1, 2) (3, 8, 1, 4) (3, 2, 4, 1) (7, 2, 3, 1) (7, 6, 2, 3) (3, 2, 7, 1) (6, 4, 5, 2) (3, 2, 0, 1) (3, 2, 9, 1) (5, 4, 6, 2) (5, 6, 4, 3) (5, 2, 8, 1) (2, 6, 7, 3) (7, 4, 6, 2) (7, 8, 5, 4) (12, 2, 3, 1) (13, 2, 3, 1) (9, 2, 3, 1) (10, 2, 3, 1) (9, 4, 1, 2) (4, 2, 1, 3) (1, 2, 4, 6) (2, 1, 5, 6) (3, 4, 1, 5) (4, 5, 1, 6) (8, 2, 1, 3) (4, 2, 3, 5) (7, 1, 2, 3) (1, 2, 3, 5) (5, 2, 1, 3) (6, 3, 1, 4) (5, 1, 3, 4) (6, 3, 2, 5) (5, 3, 1, 2) (7, 3, 2, 1) (3, 5, 4, 1) (5, 3, 2, 1) (10, 4, 1, 3) (7, 3, 1, 2) (3, 5, 1, 4) (8, 3, 1, 2) (6, 4, 1, 7) (8, 2, 3, 6) (10, 4, 3, 9) (4, 2, 1, 5) (8, 2, 3, 9) (7, 5, 1, 3) (5, 1, 2, 7) (5, 3, 1, 11) (7, 1, 3, 9) (5, 1, 2, 6) (5, 1, 2, 4) (7, 5, 1, 2) (7, 1, 3, 6) (7, 3, 2, 4) (5, 1, 2, 12) (4, 2, 1, 9) (9, 1, 4, 3) (9, 7, 1, 3) (8, 4, 2, 1) (9, 5, 2, 6) (4, 2, 1, 11) (4, 1, 9, 3) (2, 5, 9, 6) (1, 7, 9, 3) (1, 4, 6, 3) (1, 2, 4, 5) (2, 1, 5, 7) (1, 2, 4, 3) (1, 3, 5, 7) (3, 1, 7, 13) (2, 1, 5, 9) (1, 5, 7, 3) (3, 1, 7, 9) (1, 2, 4, 8) (2, 1, 5, 10) (6, 8, 2, 1) (6, 7, 1, 2) (4, 5, 1, 7) (3, 7, 4, 2) (3, 5, 2, 4) (6, 7, 1, 10) (2, 5, 3, 6) (3, 5, 2, 1) (5, 7, 2, 1) (5, 9, 4, 2) (2, 8, 6, 3) (1, 7, 6, 3) (1, 6, 5, 4) (1, 8, 7, 5) (5, 6, 1, 8) (5, 6, 1, 2) (3, 7, 4, 1) (1, 5, 4, 3)

19 (2, 7, 5, 3) (1, 4, 3, 6) (5, 7, 2, 4) (1, 7, 6, 4) (2, 9, 7, 3) (5, 7, 2, 6) (3, 4, 1, 9) (4, 7, 3, 6) (8, 10, 3, 1) (6, 4, 2, 1) (12, 2, 1, 5) (8, 6, 3, 1) (6, 7, 2, 1) (6, 5, 2, 1) (6, 5, 1, 2) (4, 1, 5, 3) (6, 2, 7, 5) (6, 4, 5, 7) (4, 6, 7, 1) (5, 1, 3, 7) (3, 1, 2, 4) (3, 4, 5, 1) (3, 5, 4, 2) (4, 2, 5, 3) (3, 7, 5, 1) (6, 5, 2, 8) (4, 3, 1, 5) (2, 4, 3, 5) (4, 1, 7, 3) (2, 4, 7, 5) (6, 2, 1, 5) (8, 2, 1, 6) (10, 3, 1, 8) (4, 1, 11, 3) (8, 2, 5, 6) (4, 1, 12, 3) (4, 1, 0, 3) (4, 1, 6, 3) (8, 1, 6, 5) (8, 3, 2, 7) (2, 6, 3, 7) (2, 4, 9, 5) (2, 6, 11, 7) (10, 1, 7, 6) (6, 1, 5, 4) (5, 7, 3, 1) (3, 11, 1, 4) (1, 9, 3, 4) (4, 6, 3, 1) (7, 9, 3, 1) (5, 7, 8, 1) (5, 9, 6, 2) (1, 11, 3, 5) (7, 11, 6, 2) (3, 7, 9, 2) (2, 8, 9, 3) (3, 5, 9, 1) (5, 3, 6, 1) (7, 5, 3, 1) (11, 1, 3, 5) (5, 3, 11, 1) (7, 3, 6, 2) (13, 7, 1, 3) (9, 5, 1, 2) (9, 1, 2, 4) (7, 1, 9, 3) (13, 1, 2, 6) (1, 4, 5, 3) (1, 4, 7, 3) (1, 4, 9, 3) (1, 4, 0, 3) (1, 4, 8, 3) (2, 6, 9, 5) (7, 2, 3, 8) (4, 2, 7, 5) (5, 2, 3, 6) (6, 3, 4, 1) (5, 0, 1, 2) (6, 1, 2, 8) (3, 1, 2, 5) (3, 7, 1, 4) (1, 5, 3, 4) (2, 1, 3, 5) (3, 9, 1, 4) (6, 3, 2, 8) (3, 12, 1, 4) (3, 0, 1, 4) (3, 6, 1, 4) (1, 2, 5, 6) (6, 3, 1, 7) (4, 2, 3, 7) (3, 6, 2, 5) (1, 2, 6, 5) (4, 3, 5, 1) (3, 6, 4, 1) (3, 9, 4, 1) (3, 0, 4, 1) (3, 8, 4, 1) (2, 4, 5, 3) (1, 2, 7, 6) (3, 6, 5, 2) (1, 8, 4, 3) (3, 2, 1, 7) (1, 2, 7, 5) (1, 3, 9, 7) (8, 1, 3, 10) (4, 1, 3, 6) (1, 2, 5, 3) (1, 3, 6, 4) (3, 4, 2, 1) (5, 6, 2, 1) (5, 4, 2, 1) (7, 2, 1, 5) (7, 3, 1, 5) (9, 3, 1, 7) (4, 5, 6, 1) (1, 6, 3, 5) (5, 7, 4, 2) (3, 5, 7, 2) (2, 5, 6, 3) (3, 4, 8, 1) (4, 3, 9, 1) (7, 1, 4, 6) (5, 3, 4, 2) (5, 2, 6, 3) (1, 3, 7, 5) (2, 1, 6, 5) (3, 1, 5, 7) (2, 1, 0, 5) (5, 1, 4, 3) (5, 3, 2, 4) (9, 3, 2, 8) (3, 1, 2, 8) (1, 3, 6, 8) (3, 1, 4, 10) (2, 1, 3, 7) (3, 9, 1, 7) (6, 2, 1, 13) (2, 4, 7, 3) (1, 2, 8, 6) (1, 3, 10, 8) (3, 6, 8, 2) (3, 6, 7, 1) (3, 9, 7, 1) (3, 9, 8, 2) (1, 9, 5, 3) (7, 1, 5, 4) (7, 5, 4, 6) (6, 3, 5, 2) (6, 9, 5, 2) (9, 3, 4, 10) (2, 1, 7, 6) (6, 3, 7, 4) (10, 5, 1, 2) (1, 2, 3, 7) (6, 2, 5, 9) (11, 1, 2, 4) (2, 1, 3, 8) (1, 0, 0, 0) (0, 1, 0, 0) (0, 0, 1, 0)

20 (0, 1, 1, 0) (2, 0, 1, 0) (3, 0, 1, 0) (1, 0, 1, 0) (1, 0, 2, 0) (1, 1, 0, 0) (1, 1, 1, 0) (2, 2, 1, 0) (1, 1, 2, 0) (1, 1, 3, 0) (1, 2, 2, 0) (4, 1, 1, 0) (2, 3, 1, 0) (1, 3, 2, 0) (1, 2, 1, 0) (3, 1, 1, 0) (2, 1, 1, 0).

Chapter 4 The asymptotic case In this chapter we consider the case t, where t = b/a and L t = O(1) p O G(4, 2 V ) (t). In this case, Lemma 5 allows us to ignore the O(1) term. More precisely, we have Lemma 6. Assume that the action of G on G(4, 2 V ) has the unique semistable orbit which is also properly stable. If t, then the semistable, properly stable and unstable loci on X are the preimages of such loci on G(4, 2 V ). Moreover, there are no properly semistable points on X. Proof: Indeed, let us denote by µ(ξ, λ) = a µ(q, λ) + b µ(π, λ) the decomposition from the beginning of Chapter 3. We may assume that λ is an element of a finite set of 1-PS listed in Chapter 3. If t, then µ(π, λ) > 0 implies µ(ξ, λ) > 0 and µ(π, λ) < 0 implies µ(ξ, λ) < 0. Because of the assumption made in the Lemma, we can assume that either of these two inequalities holds. Then we conclude that X does not contain properly semistable points (when t ) and the Lemma follows. QED

22 Let us check the assumption of Lemma 6. Lemma 7. The action of G on G(4, 2 V ) has the unique semistable orbit which is also properly stable. It corresponds to the codimension 4 linear sections of the Grassmannian which are smooth del Pezzo surfaces of degree 5. Proof: We refer the reader to Appendix A which contains the list of all codimension 4 linear sections of the Grassmannian of lines in P 4 which coincides with the list of all orbits of G = SL(5) on G(4, 2 V ). The unique open orbit gives the sections which are smooth del Pezzo surfaces. We will show that this is the only properly stable G-orbit on G(4, 2 V ) and the other orbits are unstable. Let us list destabilising 1-PS for each of the non-open orbits. In the list below we number such orbits according to their numbering in Proposition 2.1 in [14] and for each orbit provide a destabilizing 1-PS in the form (w 1, w 2, w 3, w 4, w 5 ). In each case the diagonalizing maximal torus is the standard maximal torus in SL(5) = SL(V ), where the isomorphism is given by the coordinates X 1, X 2, X 3, X 4, X 5 used by Ozeki. orbit 3: (9, 19, 11, 1, 16); orbit 4: (0, 1, 2, 1, 2); orbit 5: (1, 1, 2, 0, 2); orbit 6: ( 2, 0, 1, 1, 2); orbit 7: (1, 1, 2, 0, 2); orbit 8: (0, 2, 2, 1, 1); orbit 9: ( 2, 0, 1, 1, 2); orbit 10: (1, 0, 2, 1, 2); orbit 11: (2, 1, 1, 0, 2); orbit 12: ( 1, 2, 2, 1, 0); orbit 13: ( 1, 2, 2, 1, 0); orbit 14: (0, 2, 2, 1, 1); orbit 15: ( 1, 2, 2, 1, 0); orbit 16: ( 1, 2, 2, 1, 0); orbit 17: ( 2, 2, 1, 1, 0); orbit 18: ( 1, 2, 2, 1, 0); orbit 19: ( 1, 2, 2, 1, 0); orbit 20: ( 1, 2, 2, 1, 0); orbit 21: ( 2, 0, 2, 1, 1); orbit 22: ( 1, 2, 2, 1, 0); orbit 23: (2, 1, 2, 1, 0); orbit 24: (0, 1, 2, 2, 1); orbit 25: (0, 1, 2, 1, 2); orbit 26: ( 2, 2, 0, 1, 1); orbit 27: ( 1, 1, 2, 2, 0); orbit 28: (0, 2, 2, 1, 1); orbit 29: ( 2, 2, 1, 0, 1); orbit 30: ( 1, 2, 2, 1, 0); orbit 31: (2, 0, 1, 2, 1); orbit 32: (2, 2, 1, 0, 1); orbit 33: (2, 1, 2, 1, 0); orbit 34: (1, 2, 2, 1, 0); orbit 35: (0, 2, 2, 1, 1); orbit 36: ( 2, 2, 1, 1, 0); orbit 37: ( 2, 2, 1, 1, 0); orbit 2: (13, 8, 2, 7, 12); orbit 38: ( 1, 2, 1, 0, 2).

23 It remains to show that the open G-orbit in G(4, 2 V ) is properly stable. Each of the extremal destabilizing 1-PS above gives a set of Plücker coordinates which have a positive weight and hence have to vanish in order to lead to a point which is not properly stable. We used an Excel macros (see Appendix D) in order to identify minimal such sets and for each of them we used MATLAB in order to check that the orbit of the point in the Grassmannian for which these Plücker coordinates vanish is not general. Let us describe this process in more detail. In Appendix E we provide the MAT- LAB code which can be used in order to check that the general elements of the loci in the Grassmannian G(4, 2 V ) listed below in the form of matrices lie in the complement of the general G-orbit. Together with each locus we provide Command 2 used in the MATLAB code as well as the rank of the matrix A associated in Appendix E to a matrix representing the general element of this locus (as computed by MATLAB). 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c10 = 0, c11 = 0, c12 = 0, c20 = 0, c21 = 0, c22 = 0, c23 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0 rank = 35, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

24 MATLAB Command 2: c10 = 0, c11 = 0, c20 = 0, c21 = 0, c22 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c10 = 0, c11 = 0, c20 = 0, c21 = 0, c22 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0 rank = 38, 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c10 = 0, c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c10 = 0, c20 = 0, c21 = 0, c22 = 0, c23 = 0, c30 = 0,

25 c31 = 0, c32 = 0, c33 = 0, c34 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0 rank = 35, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c35 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c21 = 0, c22 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c21 = 0, c22 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c35 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0

26 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c21 = 0, c22 = 0, c23 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0 rank = 35, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0, c48 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0, c48 = 0 rank = 39,

27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c35 = 0, c37 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0 rank = 36, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0, c35 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0, c35 = 0, c36 = 0, c37 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0 rank = 36,

28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c20 = 0, c21 = 0, c22 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c11 = 0, c20 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0, c35 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0, c47 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c10 = 0, c12 = 0, c13 = 0, c16 = 0, c20 = 0, c21 = 0, c23 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0, c35 = 0, c36 = 0, c37 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0 rank = 36,

29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c10 = 0, c12 = 0, c13 = 0, c14 = 0, c16 = 0, c20 = 0, c21 = 0, c23 = 0, c24 = 0, c25 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0, c35 = 0, c36 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0 rank = 38, 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c10 = 0, c20 = 0, c21 = 0, c30 = 0, c31 = 0, c32 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c46 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c10 = 0, c12 = 0, c13 = 0, c14 = 0, c16 = 0, c20 = 0, c21 = 0, c22 = 0, c24 = 0, c25 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c35 = 0, c36 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0 rank = 38,

30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c13 = 0, c14 = 0, c16 = 0, c20 = 0, c21 = 0, c22 = 0, c24 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c36 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c15 = 0, c16 = 0, c20 = 0, c21 = 0, c22 = 0, c25 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c33 = 0, c34 = 0, c36 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0 rank = 39, 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 MATLAB Command 2: c11 = 0, c13 = 0, c16 = 0, c20 = 0, c23 = 0, c26 = 0, c30 = 0, c31 = 0, c32 = 0, c34 = 0, c35 = 0, c36 = 0, c37 = 0, c40 = 0, c41 = 0, c42 = 0, c43 = 0, c44 = 0, c45 = 0, c47 = 0 rank = 36.

31 The relative invariant of the prehomogeneous space which is relevant here was computed by Kimura and Sato in [10]. We used its explicit form computed as in [10] and checked (using the MATLAB code from Appendix E) that each of the loci above indeed lies in the complement of the general G-orbit (i.e. the rank of the matrix A associated in Appendix E to a matrix representing the general element of such a locus is less than 40). From the output of the Excel macros from Appendix D one sees that the following Plücker coordinates Z ijkl = Z i Z j Z k Z l, i < j < k < l of a 4-dimensional linear subspace of P( 2 V ) = P 9 with non-positive weight always vanish (i.e. their weight is always positive): Z 0123, Z 0124, Z 0125, Z 0126, Z 0127, Z 0128, Z 0129, Z 0134, Z 0135, Z 0137, Z 0145, Z 0146, Z 0147, Z 0148, Z 0149, Z 0156, Z 0157, Z 0158, Z 0159, Z 0167, Z 0234, Z 0235, Z 0237, Z 0245, Z 0246, Z 0247, Z 0248, Z 0249, Z 0256, Z 0257, Z 0258, Z 0267, Z 0345, Z 0347, Z 0357, Z 1234, Z 1235, Z 1245, Z 1246, Z 1256, Z 1345, Z 2345. Moreover, if Z 1236 does not vanish, then Z 1237 = Z 1267 = 0. If Z 1346 0, then Z 1347 = 0. If Z 1356 0, then Z 1357 = 0. If Z 2346 0, then Z 2347 = 0. Every such 4-dimensional linear subspace lies in one of the loci in G(4, 2 V ) listed above. Hence it does not lie in the general orbit of G. This shows that every element of the general G-orbit in G(4, 2 V ) is properly stable. Hence the Lemma follows. QED This proves the following theorem, which is the main result of our thesis. Theorem 1. Suppose t > 0 is large enough. Then there exists a quasi-projective quotient space X// Lt G as described in the Problem in the Introduction. Moreover, semistable and properly stable loci coincide and contain exactly double anticanonical curves on the smooth del Pezzo surface of degree 5.

Appendix A Classification of codimension 4 linear sections of the Grassmannian of lines in P 4 In this appendix we interpret a result of Ozeki [14] and give an explicit geometric classification of all codimension 4 linear sections of the Grassmannian of lines in P 4. The problem of classifying linear sections of the Grassmannian of lines in P 4 was addressed earlier by J.A. Todd [17]. We believe that our results complement his classification of codimension 4 linear sections. G = SL(5)-orbits in G(4, 2 V ) were classified algebraically by Ozeki [14]. They are listed in Proposition 2.1 in [14]. For each orbit Ozeki provides its explicit representative, so that one has explicit equations, which can be used in order to describe explicitly the corresponding linear section of the Grassmannian. We list these descriptions in Figures A.1-A.7. The (elementary) method which we use is illustrated in the case of orbit number 2. This orbit is the only orbit of codimension 1. Let us check that it can be characterized by the property that the intersection of codimension 4 linear subspaces of P 9 in this orbit with the Grassmannian G(2, 5) P 9 is a del Pezzo surface with one ordinary double point. Proposition 2.1 from [14] says that as a representative of this

33 orbit one can take equations Z 9 = 0, Z 6 = Z 1, Z 4 = Z 2, Z 5 = Z 3. If one uses equations p 1 = p 2 = p 3 = p 4 = p 5 = 0 for the Grassmannain G(2, 5) P 9, then taking intersection one obtains a surface in P 5 with coordinates Z 0, Z 1, Z 2, Z 3, Z 7, Z 8 given by the intersection of quadrics Z 0 Z 7 + Z 1 Z 3 Z 2 2, Z 0 Z 8 Z 2 Z 3 + Z 2 1, Z 2 3 Z 1 Z 2, Z 2 Z 8 Z 3 Z 7, Z 3 Z 8 Z 1 Z 7. After changing coordinates in P 5 one obtains equations shown on the figures A.1-A.7 below. It is straightforward to see that this surface S is nondegenerate irreducible and smooth except for one ordinary double point. Let us find all the lines on S. The intersection of S with hyperplane Z 5 = 0 consists of three lines and a conic intersecting at the unique point - the singular point on S. All other lines on S are also lines in its affine chart Z 5 = 1. Let us denote by X 0 = Z 0 Z 5, X 1 = Z 1 Z 5, X 2 = Z 2 Z 5, X 3 = Z 3 Z 5, X 4 = Z 4 Z 5 the coordinates in this affine chart. Then any line in this chart can be given parametrically as follows: X i = a i + b i t, i = 0, 1, 2, 3, 4, where t is a parameter and a i, b i are constants. Surface S in this affine chart is described by the equations X 3 = X 1 X 4, X 2 = X 1 X 2 4, X 0 = X 2 1X 3 4 X 2 1.

34 They give the following conditions on constants a i, b i : b 1 b 4 = 0, a 1 b 4 = 0, b 1 (a 3 4 1) = 0. If b 1 0, then we get 3 lines parametrized by η = a 4. If b 4 0, then we get one more line Z 0 = Z 1 = Z 2 = Z 3 = 0. From the well-known classification of surfaces of degree n in P n it follows that S is a del Pezzo surface of degree 5. Similarly one checks the other entries of the Figures A.1-A.7.

35 Orbit # (acc.to Ozeki [14]) Codimof Ozeki s orbit (from [14]) picture description 1 0 A smooth del Pezzo surface of degree 5. equations in with coordinates 2 1 A del Pezzo surface of degree 5 with one singular point of type. 10 lines: ( ), where 3 2 A del Pezzo surface of degree 5 with two singular points of type. 7 lines:, where 1 point: 4 2 A del Pezzo surface of degree 5 with one singular point of type. 4 lines: 2 points: 5 3 A del Pezzo surface of degree 5 with one singular point of type and one singular point of type. 4 lines: 1 point: 3 lines: 1 point: 1 point: Figure A.1. Codimension 4 sections of G(2, 5) P 9, orbits 1-5.

36 Orbit # (acc.to Ozeki [14]) Codimof Ozeki s orbit (from [14]) picture description 6 3 A del Pezzo surface of degree 5 with one singular point of type. equations in with coordinates 7 4 A nonsingular quadric surface and a nonsingular cubic scroll intersecting along a conic which is a directrix of the scroll. 2 lines: 1 point: Quadric surface: ( = ) Scroll: 8 4 A projection of a quintic scroll in (whose directrices are a conic and a twisted cubic) from a point lying in the plane generated by the conic. The projected scroll is singular along a line (the image of the conic) which is a directrix. 9 4 A del Pezzo surface of degree 5 with one singular point of type. Conic of intersection: ( = ) The second directrix of the scroll: 1 double line: The second directrix (a twisted cubic): ( = ) 10 5 A quartic scroll in and a plane intersecting along a conic which is a directrix of the scroll. 11 5 A cubic scroll and a quadric cone intersecting along a nonsingular conic which is their common directrix. 1 line: 1 point: Plane: ( = ) Conic of intersection: ( = ) The second directrix of the scroll (also a conic): Quadric cone: ( = ) Scroll: Conic of intersection: ( = ) The second directrix of the scroll: Figure A.2. Codimension 4 sections of G(2, 5) P 9, orbits 6-11.

37 Orbit # (acc.to Ozeki [14]) Codimof Ozeki s orbit (from [14]) picture description 12 5 A nonsingular quadric surface and a nonsingular cubic scroll intersecting along a pair of lines one of which is a directrix of the scroll and the other is a ruling. The second directrix of the scroll is a conic intersecting the quadric at one point. 13 5 A projection of a quintic scroll in (whose directrices are a line and a rational normal quartic) from a point lying in the plane generated by two lines a directrix and a ruling. The projected scroll is singular along a line (the image of the directrix) which is a directrix. 14 6 A plane and a Veronese surface intersecting along a conic. 15 6 A pair of nonsingular quadric surfaces and a plane. Each pair of surfaces intersects along a line. The three lines of intersection intersect at the unique point. 16 6 A plane and a nonsingular quartic scroll intersecting along a pair of lines a ruling and a directrix of the scroll. 17 6 A nonsingular quadric surface and a cone over a twisted cubic intersecting along a pair of lines two rulings. equations in Quadric surface: ( = ) Scroll: with coordinates 2 lines of intersection: (directrix) (ruling) The second directrix of the scroll (a conic): ( = ) 1 double line: The second directrix (a rational normal quartic): ( ), where t is a parameter. Plane: ( = ) Conic of intersection: ( = ) Plane: ( = ) 2 quadrics: ( = ) ( = ) Plane: ( = ) 2 lines of intersection: (directrix) (ruling) The second directrix (a twisted cubic): ( = ) Quadric surface: ( = ) Scroll: 2 lines of intersection: Vertex: ( = ) Twisted cubic: ( = ) Figure A.3. Codimension 4 sections of G(2, 5) P 9, orbits 12-17.

38 Orbit # (acc.to Ozeki [14]) Codimof Ozeki s orbit (from [14]) picture description 18 6 A two-dimensional quadric cone and a cubic scroll intersecting along two lines which are rulings of the cone a ruling and a directrix of the scoll. equations in Quadric cone: ( = ) Scroll: with coordinates 19 7 A pair of planes and a cubic scroll. Each pair of surfaces intersects along a line. The scroll intersects with one plane along a ruling and with the other plane along a directrix. The three lines intersect at one point. 2 lines of intersection: (directrix of the scroll) (ruling of the scroll) The second directrix of the scroll (a conic): ( = ) 2 planes: ( = ) ( = ) Scroll: 20 7 A nonsingular quadric surface, a quadric cone and a plane. Each pair of surfaces intersects along a line. Three lines intersect at one point which is the vertex of the cone. 2 lines of intersection of the scroll with planes: (directrix) (ruling) The second directrix of the scroll (a conic): ( = ) 2 quadrics: ( = ) ( = ) Plane: 21 7 A quadric cone and a cone over a twisted cubic intersecting along a common ruling. The line of intersection passes through vertices of both cones. 22 8 A plane (with multiplicity 1) and a nonsingular quadric surface (with multiplicity 2) intersecting along a line. Quadric cone: ( = ) Twisted cubic: ( = = ) Vertex of the cubic cone: 1 line of intersection: Quadric: ( = ) Plane: Figure A.4. Codimension 4 sections of G(2, 5) P 9, orbits 18-22.