surfaces_and_arcs.h

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/*
 * surfaces_and_arcs.h
 *
 *  Created on: Jun 26, 2021
 *      Author: betten
 */
 
#ifndef SRC_LIB_TOP_LEVEL_SURFACES_SURFACES_AND_ARCS_SURFACES_AND_ARCS_H_
#define SRC_LIB_TOP_LEVEL_SURFACES_SURFACES_AND_ARCS_SURFACES_AND_ARCS_H_
 
namespace orbiter {
namespace layer5_applications {
namespace applications_in_algebraic_geometry {
namespace cubic_surfaces_and_arcs {
 
 
// #############################################################################
// arc_lifting.cpp
// #############################################################################
 
 
 
class arc_lifting {
 
public:
 
    int q;
    field_theory::finite_field *F;
 
    algebraic_geometry::surface_domain *Surf;
 
    cubic_surfaces_in_general::surface_with_action *Surf_A;
 
    long int *arc;
    int arc_size;
 
 
 
 
 
    int *the_equation; // [20]
 
    algebraic_geometry::web_of_cubic_curves *Web;
 
 
    trihedral_pair_with_action *Trihedral_pair;
 
    arc_lifting();
    ~arc_lifting();
    void null();
    void freeself();
    void create_surface_and_group(
            cubic_surfaces_in_general::surface_with_action *Surf_A, long int *Arc6,
        int verbose_level);
    void create_web_of_cubic_curves(int verbose_level);
    void report(std::ostream &ost, int verbose_level);
    void report_equation(std::ostream &ost);
};
 
 
// #############################################################################
// arc_orbits_on_pairs.cpp
// #############################################################################
 
 
 
class arc_orbits_on_pairs {
public:
 
    surfaces_arc_lifting *SAL;
 
    actions::action *A; // this is the 3x3 group
 
    data_structures_groups::set_and_stabilizer *The_arc;
    actions::action *A_on_arc;
 
    int arc_idx;
    poset_classification::poset_with_group_action *Poset;
    poset_classification::poset_classification_control *Control;
    poset_classification::poset_classification *Orbits_on_pairs;
 
    int nb_orbits_on_pairs;
 
    arc_partition *Table_orbits_on_partition; // [nb_orbits_on_pairs]
 
    int total_nb_orbits_on_partitions;
 
    int *partition_orbit_first; // [nb_orbits_on_pairs]
    int *partition_orbit_len; // [nb_orbits_on_pairs]
 
 
    arc_orbits_on_pairs();
    ~arc_orbits_on_pairs();
    void null();
    void freeself();
    void init(
        surfaces_arc_lifting *SAL, int arc_idx,
        actions::action *A,
        int verbose_level);
    void print();
    void recognize(long int *pair, int *transporter,
            int &orbit_idx, int verbose_level);
 
};
 
 
// #############################################################################
// arc_partition.cpp
// #############################################################################
 
 
 
class arc_partition {
public:
 
    arc_orbits_on_pairs *OP;
 
    actions::action *A;
    actions::action *A_on_arc;
 
    int pair_orbit_idx;
    data_structures_groups::set_and_stabilizer *The_pair;
 
    long int arc_remainder[4];
 
    actions::action *A_on_rest;
    actions::action *A_on_partition;
 
    groups::schreier *Orbits_on_partition;
 
    int nb_orbits_on_partition;
 
 
    arc_partition();
    ~arc_partition();
    void null();
    void freeself();
    void init(
        arc_orbits_on_pairs *OP, int pair_orbit_idx,
        actions::action *A, actions::action *A_on_arc,
        int verbose_level);
    void recognize(int *partition, int *transporter,
            int &orbit_idx, int verbose_level);
 
};
 
 
// #############################################################################
// classify_trihedral_pairs.cpp
// #############################################################################
 
 
 
 
class classify_trihedral_pairs {
 
public:
 
    int q;
    field_theory::finite_field *F; // do not free
    actions::action *A; // do not free
 
    cubic_surfaces_in_general::surface_with_action *Surf_A; // do not free
    algebraic_geometry::surface_domain *Surf; // do not free
 
    groups::strong_generators *gens_type1;
    groups::strong_generators *gens_type2;
 
    poset_classification::poset_with_group_action *Poset1;
    poset_classification::poset_with_group_action *Poset2;
    poset_classification::poset_classification *orbits_on_trihedra_type1;
    poset_classification::poset_classification *orbits_on_trihedra_type2;
 
    int nb_orbits_type1;
    int nb_orbits_type2;
    int nb_orbits_ordered_total;
 
    invariant_relations::flag_orbits *Flag_orbits;
 
    int nb_orbits_trihedral_pairs;
 
    invariant_relations::classification_step *Trihedral_pairs;
 
 
 
    classify_trihedral_pairs();
    ~classify_trihedral_pairs();
    void null();
    void freeself();
    void init(cubic_surfaces_in_general::surface_with_action *Surf_A,
            int verbose_level);
 
    void classify_orbits_on_trihedra(
            poset_classification::poset_classification_control *Control1,
            poset_classification::poset_classification_control *Control2,
            int verbose_level);
    void report_summary(std::ostream &ost);
    void report(std::ostream &ost);
    void list_orbits_on_trihedra_type1(std::ostream &ost, int f_detailed);
    void list_orbits_on_trihedra_type2(std::ostream &ost, int f_detailed);
    void early_test_func_type1(long int *S, int len,
            long int *candidates, int nb_candidates,
            long int *good_candidates, int &nb_good_candidates,
        int verbose_level);
    void early_test_func_type2(long int *S, int len,
            long int *candidates, int nb_candidates,
            long int *good_candidates, int &nb_good_candidates,
        int verbose_level);
    void identify_three_planes(int p1, int p2, int p3,
        int &type, int *transporter, int verbose_level);
    void classify(
            poset_classification::poset_classification_control *Control1,
            poset_classification::poset_classification_control *Control2,
            int verbose_level);
    void downstep(int verbose_level);
    void upstep(int verbose_level);
    void print_trihedral_pairs_summary(std::ostream &ost);
    void print_trihedral_pairs(std::ostream &ost,
        int f_with_stabilizers);
    groups::strong_generators *identify_trihedral_pair_and_get_stabilizer(
        long int *planes6, int *transporter, int &orbit_index,
        int verbose_level);
    void identify_trihedral_pair(long int *planes6,
        int *transporter, int &orbit_index, int verbose_level);
 
};
 
 
 
 
// #############################################################################
// six_arcs_not_on_a_conic.cpp
// #############################################################################
 
 
 
class six_arcs_not_on_a_conic {
 
public:
 
    //projective_space *P2; // do not free
    apps_geometry::arc_generator_description *Descr;
    projective_geometry::projective_space_with_action *PA;
 
    apps_geometry::arc_generator *Gen;
 
    int nb_orbits;
 
    int *Not_on_conic_idx;
    int nb_arcs_not_on_conic;
 
    six_arcs_not_on_a_conic();
    ~six_arcs_not_on_a_conic();
    void null();
    void freeself();
    void init(
            apps_geometry::arc_generator_description *Descr,
            projective_geometry::projective_space_with_action *PA,
        int f_test_nb_Eckardt_points, int nb_E,
        //algebraic_geometry::surface_domain *Surf,
        int verbose_level);
    void recognize(long int *arc6, int *transporter,
            int &orbit_not_on_conic_idx, int verbose_level);
    void report_latex(std::ostream &ost);
    void report_specific_arc_basic(std::ostream &ost, int arc_idx);
    void report_specific_arc(std::ostream &ost, int arc_idx);
};
 
 
 
// #############################################################################
// surface_classify_using_arc.cpp
// #############################################################################
 
 
 
class surface_classify_using_arc {
public:
 
    cubic_surfaces_in_general::surface_with_action *Surf_A;
 
 
 
    six_arcs_not_on_a_conic *Six_arcs;
    apps_geometry::arc_generator_description *Descr;
 
    int *transporter;
 
 
    int nb_surfaces;
    surface_create_by_arc_lifting *SCAL;
    // allocated as [Six_arcs->nb_arcs_not_on_conic], used as [nb_surfaces]
 
    int *Arc_identify_nb;
    int *Arc_identify;  // [Six_arcs->nb_arcs_not_on_conic]
    //[Six_arcs->nb_arcs_not_on_conic * Six_arcs->nb_arcs_not_on_conic]
    int *f_deleted; // [Six_arcs->nb_arcs_not_on_conic]
 
    int *Decomp; // [Six_arcs->nb_arcs_not_on_conic * nb_surfaces]
 
    surface_classify_using_arc();
    ~surface_classify_using_arc();
    void classify_surfaces_through_arcs_and_trihedral_pairs(
            poset_classification::poset_classification_control *Control_six_arcs,
            cubic_surfaces_in_general::surface_with_action *Surf_A,
            int f_test_nb_Eckardt_points, int nb_E,
            int verbose_level);
    void report(
            graphics::layered_graph_draw_options *Opt,
            int verbose_level);
    void report2(std::ostream &ost,
            graphics::layered_graph_draw_options *Opt,
            int verbose_level);
    void report_decomposition_matrix(std::ostream &ost, int verbose_level);
};
 
// #############################################################################
// surface_create_by_arc_lifting.cpp
// #############################################################################
 
 
 
class surface_create_by_arc_lifting {
public:
 
    surface_classify_using_arc *SCA;
 
    int arc_idx;
    int surface_idx;
    long int *Arc6;
 
 
    arc_lifting *AL;
 
    cubic_surfaces_in_general::surface_object_with_action *SOA;
 
 
    int nine_lines_idx[9];
    std::string arc_label;
    std::string arc_label_short;
 
    cubic_surfaces_in_general::surface_clebsch_map *Clebsch; // [SOA->Orbits_on_single_sixes->nb_orbits]
    int *Other_arc_idx; // [SOA->Orbits_on_single_sixes->nb_orbits]
 
    surface_create_by_arc_lifting();
    ~surface_create_by_arc_lifting();
    void init(int arc_idx,
            surface_classify_using_arc *SCA, int verbose_level);
    void report_summary(std::ostream &ost, int verbose_level);
    void report(std::ostream &ost,
            graphics::layered_graph_draw_options *Opt,
            int verbose_level);
 
};
 
 
// #############################################################################
// surfaces_arc_lifting_definition_node.cpp
// #############################################################################
 
 
 
 
 
class surfaces_arc_lifting_definition_node {
public:
 
    surfaces_arc_lifting *Lift;
 
    int f;
    int orbit_idx;
 
    algebraic_geometry::surface_object *SO;
    cubic_surfaces_in_general::surface_object_with_action *SOA;
 
 
 
    groups::strong_generators *Flag_stab_gens;
    ring_theory::longinteger_object Flag_stab_go;
 
 
    int three_lines_idx[45 * 3];
        // the index into Lines[] of the
        // three lines in the chosen tritangent plane
        // This is computed from the Schlaefli labeling
        // using the eckardt point class.
 
    long int three_lines[45 * 3];
        // the three lines in the chosen tritangent plane
 
 
 
    algebraic_geometry::seventytwo_cases Seventytwo[45 * 72];
        // for each of the 45 tritangent planes,
        // there are 72 Clebsch maps
 
 
    int nb_coset_reps;
    surfaces_arc_lifting_trace **T; // [nb_coset_reps]
    data_structures_groups::vector_ge *coset_reps;
 
    int *relative_order_table; // [nb_coset_reps]
 
    int f_has_F2;
    int *F2; // F2[i] = Seventytwo[i].f2;
    data_structures::tally *tally_F2;
 
 
    surfaces_arc_lifting_definition_node();
    ~surfaces_arc_lifting_definition_node();
    void init_with_27_lines(surfaces_arc_lifting *Lift,
            int f, int orbit_idx, long int *Lines27, int *eqn20,
            int verbose_level);
    void tally_f2(int verbose_level);
    void report(int verbose_level);
    void report2(std::ostream &ost, int verbose_level);
    void report_cosets(std::ostream &ost, int verbose_level);
    void report_cosets_detailed(std::ostream &ost, int verbose_level);
    void report_cosets_HDS(std::ostream &ost, int verbose_level);
    void report_HDS_top(std::ostream &ost);
    void report_HDS_bottom(std::ostream &ost);
    void report_cosets_T3(std::ostream &ost, int verbose_level);
    void report_T3_top(std::ostream &ost);
    void report_T3_bottom(std::ostream &ost);
    void report_tally_F2(std::ostream &ost, int verbose_level);
    void report_Clebsch_maps(std::ostream &ost, int verbose_level);
    void report_Clebsch_maps_for_one_tritangent_plane(std::ostream &ost,
            int plane_idx, int verbose_level);
};
 
 
// #############################################################################
// surfaces_arc_lifting_trace.cpp
// #############################################################################
 
 
 
 
 
class surfaces_arc_lifting_trace {
public:
 
    surfaces_arc_lifting_upstep *Up;
 
    int f, f2;
 
    int po, so;
 
    // po = Lift->Flag_orbits->Flag_orbit_node[f].downstep_primary_orbit;
    // so = Lift->Flag_orbits->Flag_orbit_node[f].downstep_secondary_orbit;
 
    // 3x3 matrices of elements in PGGL(3,q)
    int *Elt_alpha2;
        // Using local coordinates P6_local[6],
        // maps the arc P6[6] to the canonical arc in the classification.
    int *Elt_beta1;
        // Moves the arc points on m1 to P1 and P2.
        // Computed using
        // Table_orbits_on_pairs[orbit_not_on_conic_idx].recognize
    int *Elt_beta2;
        // Moves the partition, computed using
        // Table_orbits_on_partition[pair_orbit_idx].recognize
 
    // temporary matrices
    int *Elt_T1;
    int *Elt_T2;
    int *Elt_T3;
    int *Elt_T4;
 
    // 4x4 matrices or elements in PGGL(4,q):
    int *Elt_Alpha1;
        // Moves the chosen tritangent plane to the hyperplane X3=0
 
    int *Elt_Alpha2; // embedding of alpha2
    int *Elt_Beta1; // embedding of beta1
    int *Elt_Beta2; // embedding of beta2
    int *Elt_Beta3;
        // Moves the two lines  which are the images of l1 and l2
        // under the group elements computed so far
        // to the canonical ones associated with the flag f2.
        // Computed with hyperplane_lifting_with_two_lines_moved
 
        // if f2 = f then
        // Alpha1 * Alpha2 * Beta1 * Beta2 * Beta3
        // is an automorphism of the surface
 
 
    int upstep_idx;
 
 
 
    int seventytwo_case_idx;
 
    algebraic_geometry::seventytwo_cases The_case;
 
 
    surfaces_arc_lifting_trace();
    ~surfaces_arc_lifting_trace();
    void init(surfaces_arc_lifting_upstep *Up,
            int seventytwo_case_idx, int verbose_level);
    void process_flag_orbit(surfaces_arc_lifting_upstep *Up, int verbose_level);
    void move_arc(int verbose_level);
    void move_plane_and_arc(long int *P6a, int verbose_level);
    void make_arc_canonical(
            long int *P6_local, long int *P6_local_canonical,
            int &orbit_not_on_conic_idx, int verbose_level);
    void compute_beta1(algebraic_geometry::seventytwo_cases *The_case, int verbose_level);
    void compute_beta2(int orbit_not_on_conic_idx,
            int pair_orbit_idx, int &partition_orbit_idx,
            int *the_partition4, int verbose_level);
    void lift_group_elements_and_move_two_lines(int verbose_level);
    void embed(int *Elt_A3, int *Elt_A4, int verbose_level);
    void report_product(std::ostream &ost, int *Elt, int verbose_level);
 
};
 
 
 
// #############################################################################
// surfaces_arc_lifting_upstep.cpp
// #############################################################################
 
 
 
 
 
 
class surfaces_arc_lifting_upstep {
public:
 
    surfaces_arc_lifting *Lift;
 
    int *f_processed; // [Lift->Flag_orbits->nb_flag_orbits]
    int nb_processed;
 
    int pt_representation_sz;
    long int *Flag_representation;
    long int *Flag2_representation;
        // used only in upstep_group_elements
 
    ring_theory::longinteger_object A4_go;
 
 
    double progress;
    long int Lines[27];
    int eqn20[20];
 
 
    surfaces_arc_lifting_definition_node *D;
 
    //vector_ge *coset_reps;
    //int nb_coset_reps;
 
    //strong_generators *Flag_stab_gens;
    //longinteger_object Flag_stab_go;
 
    int f;
 
    int tritangent_plane_idx;
        // the tritangent plane picked for the Clebsch map,
        // using the Schlaefli labeling, in [0,44].
 
 
    int three_lines_idx[3];
        // the index into Lines[] of the
        // three lines in the chosen tritangent plane
        // This is computed from the Schlaefli labeling
        // using the eckardt point class.
 
    long int three_lines[3];
        // the three lines in the chosen tritangent plane
 
 
 
    algebraic_geometry::seventytwo_cases Seventytwo[72];
 
    int seventytwo_case_idx;
 
 
    surfaces_arc_lifting_upstep();
    ~surfaces_arc_lifting_upstep();
    void init(surfaces_arc_lifting *Lift, int verbose_level);
    void process_flag_orbit(int verbose_level);
    void compute_stabilizer(surfaces_arc_lifting_definition_node *D,
            groups::strong_generators *&Aut_gens, int verbose_level);
    void process_tritangent_plane(surfaces_arc_lifting_definition_node *D,
            int verbose_level);
    void make_seventytwo_cases(int verbose_level);
 
};
 
 
 
 
 
// #############################################################################
// surfaces_arc_lifting.cpp
// #############################################################################
 
 
 
class surfaces_arc_lifting {
public:
    field_theory::finite_field *F;
    int q;
    groups::linear_group *LG4; // PGL(4,q)
 
    int f_semilinear;
 
    std::string fname_base;
 
    actions::action *A4; // the action of PGL(4,q) on points
    actions::action *A3; // the action of PGL(3,q) on points
 
    algebraic_geometry::surface_domain *Surf;
    cubic_surfaces_in_general::surface_with_action *Surf_A;
 
    six_arcs_not_on_a_conic *Six_arcs;
 
    arc_orbits_on_pairs *Table_orbits_on_pairs;
        // [Six_arcs->nb_arcs_not_on_conic]
 
    int nb_flag_orbits;
 
    // classification of surfaces:
    invariant_relations::flag_orbits *Flag_orbits;
 
    int *flag_orbit_fst; // [Six_arcs->nb_arcs_not_on_conic]
    int *flag_orbit_len; // [Six_arcs->nb_arcs_not_on_conic]
 
    int *flag_orbit_on_arcs_not_on_a_conic_idx; // [Flag_orbits->nb_flag_orbits]
    int *flag_orbit_on_pairs_idx; // [Flag_orbits->nb_flag_orbits]
    int *flag_orbit_on_partition_idx; // [Flag_orbits->nb_flag_orbits]
 
    invariant_relations::classification_step *Surfaces;
 
    surfaces_arc_lifting();
    ~surfaces_arc_lifting();
    void null();
    void freeself();
    void init(
            cubic_surfaces_in_general::surface_with_action *Surf_A,
        poset_classification::poset_classification_control *Control_six_arcs,
        int f_test_nb_Eckardt_points, int nb_E,
        int verbose_level);
    void downstep(int verbose_level);
    void downstep_one_arc(int arc_idx,
            int &cur_flag_orbit, long int *Flag, int verbose_level);
    void report(
            graphics::layered_graph_draw_options *draw_options,
            int verbose_level);
    void report2(std::ostream &ost,
            graphics::layered_graph_draw_options *draw_options,
            int verbose_level);
    void report_flag_orbits(std::ostream &ost, int verbose_level);
    void report_flag_orbits_in_detail(std::ostream &ost, int verbose_level);
    void report_surfaces_in_detail(std::ostream &ost, int verbose_level);
};
 
 
 
// #############################################################################
// trihedral_pair_with_action.cpp
// #############################################################################
 
 
 
class trihedral_pair_with_action {
 
public:
 
    arc_lifting *AL;
 
    int The_six_plane_equations[6 * 4]; // [6 * 4]
    int *The_surface_equations; // [(q + 1) * 20]
    long int plane6_by_dual_ranks[6];
    int lambda, lambda_rk;
    int t_idx;
 
    groups::strong_generators *stab_gens_trihedral_pair; // stabilizer of trihedral pair
    groups::strong_generators *gens_subgroup;
    ring_theory::longinteger_object stabilizer_of_trihedral_pair_go;
    actions::action *A_on_equations;
    groups::schreier *Orb;
    ring_theory::longinteger_object stab_order;
    int trihedral_pair_orbit_index;
    data_structures_groups::vector_ge *cosets;
 
    data_structures_groups::vector_ge *coset_reps;
    long int nine_lines[9];
    int *aut_T_index;
    int *aut_coset_index;
    groups::strong_generators *Aut_gens;
 
 
    int F_plane[3 * 4];
    int G_plane[3 * 4];
    int *System; // [3 * 4 * 3]
    //int nine_lines[9];
 
    int Iso_type_as_double_triplet[120];
    data_structures::tally *Double_triplet_type_distribution;
    data_structures::set_of_sets *Double_triplet_types;
    int *Double_triplet_type_values;
    int nb_double_triplet_types;
 
    int *transporter0;
    int *transporter;
    int *Elt1;
    int *Elt2;
    int *Elt3;
    int *Elt4;
    int *Elt5;
 
 
    trihedral_pair_with_action();
    ~trihedral_pair_with_action();
    void init(arc_lifting *AL, int verbose_level);
    void loop_over_trihedral_pairs(
            data_structures_groups::vector_ge *cosets,
            data_structures_groups::vector_ge *&coset_reps,
        int *&aut_T_index, int *&aut_coset_index, int verbose_level);
    void create_the_six_plane_equations(int t_idx,
        int verbose_level);
    void create_surface_from_trihedral_pair_and_arc(
        int t_idx,
        int verbose_level);
        // plane6[6]
        // The_six_plane_equations[6 * 4]
        // The_surface_equations[(q + 1) * 20]
    groups::strong_generators *create_stabilizer_of_trihedral_pair(
            int &trihedral_pair_orbit_index, int verbose_level);
    void create_action_on_equations_and_compute_orbits(
        int *The_surface_equations,
        groups::strong_generators *gens_for_stabilizer_of_trihedral_pair,
        actions::action *&A_on_equations, groups::schreier *&Orb,
        int verbose_level);
    void create_clebsch_system(int verbose_level);
    void compute_iso_types_as_double_triplets(int verbose_level);
    void print_FG(std::ostream &ost);
    void print_equations();
    void report(std::ostream &ost, int verbose_level);
    void report_iso_type_as_double_triplets(std::ostream &ost);
 
};
 
 
 
 
}}}}
 
 
 
 
 
 
#endif /* SRC_LIB_TOP_LEVEL_SURFACES_SURFACES_AND_ARCS_SURFACES_AND_ARCS_H_ */