Changeset 11d225 in git

Timestamp:

Nov 6, 2013, 4:34:15 PM (10 years ago)

Author:

Hans Schoenemann <hannes@…>

Branches:

(u'spielwiese', 'd1b01e9d51ade4b46b745d3bada5c5f3696be3a8')

Children:

2ea781ea236dc6d65399e46b262e9954538749a1

Parents:

da7eaf561933d677968dbf5e5b17b8a9e33117d5

Message:

chg: new version of schubert.lib

File:

• Singular/LIB/schubert.lib (modified) (38 diffs)

Singular/LIB/schubert.lib

@@ -1 +1 @@
 ////////////////////////////////////////////////////////////////////////////////
-version="version schubert.lib 4.0.0.0 Jun_2013 "; // $Id$
+version="version schubert.lib 4.0.0.0 Nov_2013 "; // $Id$
 category="Algebraic Geometry";
 info="
-LIBRARY:    Schubert.lib    Proceduces for Intersection Theory
-
-AUTHOR:     Hiep Dang,      email: hiep@mathematik.uni-kl.de
+LIBRARY:    schubert.lib    Proceduces for Intersection Theory
+
+AUTHOR:     Hiep Dang,          email: hiep@mathematik.uni-kl.de
 
 OVERVIEW:
 
     We implement new classes (variety, sheaf, stack, graph) and methods for
-    computing with them. Here a variety is represented by a nonnegative integer
-    which is its dimension and a graded ring which is its Chow ring. A sheaf is
-    represented by a variety and a polynomial which is its Chern character.
-    In particular, we implement the concrete varieties such as projective spaces
-    , Grassmannians, and projective bundles.
+    computing with them. An abstract variety is represented by a nonnegative
+    integer which is its dimension and a graded ring which is its Chow ring.
+    An abstract sheaf is represented by a variety and a polynomial which is its
+    Chern character. In particular, we implement the concrete varieties such as
+    projective spaces, Grassmannians, and projective bundles.
 
     An important task of this library is related to the computation of
@@ -24 +24 @@
     Kontsevich, and the graphs corresponding to the fixed point components of a
     torus action on the moduli spaces of stable maps.
+
+    As an insightful example, the numbers of rational curves on general complete
+    intersection Calabi-Yau threefolds in projective spaces are computed up to
+    degree 6. The results are all in agreement with predictions made from mirror
+    symmetry computations.
 
 REFERENCES:
@@ -95 +100 @@
     dualPartition(list)                 compute the dual of a partition
 
-KEYWORDS:       Intersection Theory; Enumerative Geometry; Schubert Calculus;
-                Bott's formula.
+KEYWORDS:       Intersection theory; Enumerative geometry; Schubert calculus;
+                Bott's formula; Gromov-Witten invariants.
 
 ";
@@ -217 +222 @@
 "
 {
-    int i,j,k,h,m,n,p;
+    int i,j,k,h,m,n,p,q;
     list l;
     int d = M.degreeCurve;
@@ -228 +233 @@
             if (i <> j)
             {
-                l = insert(l,list(graph1(d,i,j),2*d),size(l));
+                l[size(l)+1] = list(graph1(d,i,j),2*d);
             }
         }
@@ -242 +247 @@
                     if (i <> j and j <> k)
                     {
-                        l = insert(l,list(graph2(list(1,1),i,j,k),2),size(l));
+                        l[size(l)+1] = list(graph2(list(1,1),i,j,k),2);
                     }
                 }
@@ -258 +263 @@
                     if (i <> j and j <> k)
                     {
-                        l = insert(l,list(graph2(list(2,1),i,j,k),2),size(l));
+                        l[size(l)+1] = list(graph2(list(2,1),i,j,k),2);
                         for (h=0;h<=r;h++)
                         {
                             if (h <> k)
                             {
-                                l = insert(l,list(graph31(list(1,1,1),i,j,k,h),2),size(l));
+                                l[size(l)+1] = list(graph31(list(1,1,1),i,j,k,h),2);
                             }
                             if (h <> j)
                             {
-                                l = insert(l,list(graph32(list(1,1,1),i,j,k,h),6),size(l));
+                                l[size(l)+1] = list(graph32(list(1,1,1),i,j,k,h),6);
                             }
                         }
@@ -285 +290 @@
                     if (i <> j and j <> k)
                     {
-                        l = insert(l,list(graph2(list(3,1),i,j,k),3),size(l));
-                        l = insert(l,list(graph2(list(2,2),i,j,k),8),size(l));
+                        l[size(l)+1] = list(graph2(list(3,1),i,j,k),3);
+                        l[size(l)+1] = list(graph2(list(2,2),i,j,k),8);
                         for (h=0;h<=r;h++)
                         {
                             if (h <> k)
                             {
-                                l = insert(l,list(graph31(list(2,1,1),i,j,k,h),2),size(l));
-                                l = insert(l,list(graph31(list(1,2,1),i,j,k,h),4),size(l));
+                                l[size(l)+1] = list(graph31(list(2,1,1),i,j,k,h),2);
+                                l[size(l)+1] = list(graph31(list(1,2,1),i,j,k,h),4);
                             }
                             if (h <> j)
                             {
-                                l = insert(l,list(graph32(list(2,1,1),i,j,k,h),4),size(l));
+                                l[size(l)+1] = list(graph32(list(2,1,1),i,j,k,h),4);
                             }
                             for (m=0;m<=r;m++)
@@ -302 +307 @@
                                 if (k <> h and m <> h)
                                 {
-                                    l = insert(l,list(graph41(list(1,1,1,1),i,j,k,h,m),2),size(l));
+                                    l[size(l)+1] = list(graph41(list(1,1,1,1),i,j,k,h,m),2);
                                 }
                                 if (k <> h and m <> k)
                                 {
-                                    l = insert(l,list(graph42(list(1,1,1,1),i,j,k,h,m),2),size(l));
+                                    l[size(l)+1] = list(graph42(list(1,1,1,1),i,j,k,h,m),2);
                                 }
                                 if (h <> j and m <> j)
                                 {
-                                    l = insert(l,list(graph43(list(1,1,1,1),i,j,k,h,m),24),size(l));
+                                    l[size(l)+1] = list(graph43(list(1,1,1,1),i,j,k,h,m),24);
                                 }
                             }
@@ -329 +334 @@
                     if (i <> j and j <> k)
                     {
-                        l = insert(l,list(graph2(list(4,1),i,j,k),4),size(l));
-                        l = insert(l,list(graph2(list(3,2),i,j,k),6),size(l));
+                        l[size(l)+1] = list(graph2(list(4,1),i,j,k),4);
+                        l[size(l)+1] = list(graph2(list(3,2),i,j,k),6);
                         for (h=0;h<=r;h++)
                         {
                             if (k <> h)
                             {
-                                l = insert(l,list(graph31(list(3,1,1),i,j,k,h),3),size(l));
-                                l = insert(l,list(graph31(list(1,3,1),i,j,k,h),6),size(l));
-                                l = insert(l,list(graph31(list(2,2,1),i,j,k,h),4),size(l));
-                                l = insert(l,list(graph31(list(2,1,2),i,j,k,h),8),size(l));
+                                l[size(l)+1] = list(graph31(list(3,1,1),i,j,k,h),3);
+                                l[size(l)+1] = list(graph31(list(1,3,1),i,j,k,h),6);
+                                l[size(l)+1] = list(graph31(list(2,2,1),i,j,k,h),4);
+                                l[size(l)+1] = list(graph31(list(2,1,2),i,j,k,h),8);
                             }
                             if (j <> h)
                             {
-                                l = insert(l,list(graph32(list(3,1,1),i,j,k,h),6),size(l));
-                                l = insert(l,list(graph32(list(2,2,1),i,j,k,h),8),size(l));
+                                l[size(l)+1] = list(graph32(list(3,1,1),i,j,k,h),6);
+                                l[size(l)+1] = list(graph32(list(2,2,1),i,j,k,h),8);
                             }
                             for (m=0;m<=r;m++)
@@ -349 +354 @@
                                 if (k <> h and h <> m)
                                 {
-                                    l = insert(l,list(graph41(list(2,1,1,1),i,j,k,h,m),2),size(l));
-                                    l = insert(l,list(graph41(list(1,2,1,1),i,j,k,h,m),2),size(l));
+                                    l[size(l)+1] = list(graph41(list(2,1,1,1),i,j,k,h,m),2);
+                                    l[size(l)+1] = list(graph41(list(1,2,1,1),i,j,k,h,m),2);
                                 }
                                 if (k <> h and k <> m)
                                 {
-                                    l = insert(l,list(graph42(list(2,1,1,1),i,j,k,h,m),4),size(l));
-                                    l = insert(l,list(graph42(list(1,2,1,1),i,j,k,h,m),4),size(l));
-                                    l = insert(l,list(graph42(list(1,1,2,1),i,j,k,h,m),2),size(l));
+                                    l[size(l)+1] = list(graph42(list(2,1,1,1),i,j,k,h,m),4);
+                                    l[size(l)+1] = list(graph42(list(1,2,1,1),i,j,k,h,m),4);
+                                    l[size(l)+1] = list(graph42(list(1,1,2,1),i,j,k,h,m),2);
                                 }
                                 if (j <> h and j <> m)
                                 {
-                                    l = insert(l,list(graph43(list(2,1,1,1),i,j,k,h,m),12),size(l));
+                                    l[size(l)+1] = list(graph43(list(2,1,1,1),i,j,k,h,m),12);
                                 }
                                 for (n=0;n<=r;n++)
@@ -366 +371 @@
                                     if (k <> h and h <> m and m <> n)
                                     {
-                                        l = insert(l,list(graph51(list(1,1,1,1,1),i,j,k,h,m,n),2),size(l));
+                                        l[size(l)+1] = list(graph51(list(1,1,1,1,1),i,j,k,h,m,n),2);
                                     }
                                     if (k <> h and h <> m and h <> n)
                                     {
-                                        l = insert(l,list(graph52(list(1,1,1,1,1),i,j,k,h,m,n),2),size(l));
+                                        l[size(l)+1] = list(graph52(list(1,1,1,1,1),i,j,k,h,m,n),2);
                                     }
                                     if (k <> h and k <> m and k <> n)
                                     {
-                                        l = insert(l,list(graph53(list(1,1,1,1,1),i,j,k,h,m,n),6),size(l));
+                                        l[size(l)+1] = list(graph53(list(1,1,1,1,1),i,j,k,h,m,n),6);
                                     }
                                     if (j <> h and h <> m and h <> n)
                                     {
-                                        l = insert(l,list(graph54(list(1,1,1,1,1),i,j,k,h,m,n),8),size(l));
+                                        l[size(l)+1] = list(graph54(list(1,1,1,1,1),i,j,k,h,m,n),8);
                                     }
                                     if (k <> h and k <> m and h <> n)
                                     {
-                                        l = insert(l,list(graph55(list(1,1,1,1,1),i,j,k,h,m,n),2),size(l));
+                                        l[size(l)+1] = list(graph55(list(1,1,1,1,1),i,j,k,h,m,n),2);
                                     }
                                     if (j <> h and j <> m and j <> n)
                                     {
-                                        l = insert(l,list(graph56(list(1,1,1,1,1),i,j,k,h,m,n),120),size(l));
+                                        l[size(l)+1] = list(graph56(list(1,1,1,1,1),i,j,k,h,m,n),120);
                                     }
                                 }
@@ -406 +411 @@
                     if (i <> j and j <> k)
                     {
-                        l = insert(l,list(graph2(list(5,1),i,j,k),5),size(l));
-                        l = insert(l,list(graph2(list(4,2),i,j,k),8),size(l));
-                        l = insert(l,list(graph2(list(3,3),i,j,k),18),size(l));
+                        l[size(l)+1] = list(graph2(list(5,1),i,j,k),5);
+                        l[size(l)+1] = list(graph2(list(4,2),i,j,k),8);
+                        l[size(l)+1] = list(graph2(list(3,3),i,j,k),18);
                         for (h=0;h<=r;h++)
                         {
                             if (k <> h)
                             {
-                                l = insert(l,list(graph31(list(4,1,1),i,j,k,h),4),size(l));
-                                l = insert(l,list(graph31(list(1,4,1),i,j,k,h),8),size(l));
-                                l = insert(l,list(graph31(list(3,2,1),i,j,k,h),6),size(l));
-                                l = insert(l,list(graph31(list(3,1,2),i,j,k,h),6),size(l));
-                                l = insert(l,list(graph31(list(1,3,2),i,j,k,h),6),size(l));
-                                l = insert(l,list(graph31(list(2,2,2),i,j,k,h),16),size(l));
+                                l[size(l)+1] = list(graph31(list(4,1,1),i,j,k,h),4);
+                                l[size(l)+1] = list(graph31(list(1,4,1),i,j,k,h),8);
+                                l[size(l)+1] = list(graph31(list(3,2,1),i,j,k,h),6);
+                                l[size(l)+1] = list(graph31(list(3,1,2),i,j,k,h),6);
+                                l[size(l)+1] = list(graph31(list(1,3,2),i,j,k,h),6);
+                                l[size(l)+1] = list(graph31(list(2,2,2),i,j,k,h),16);
                             }
                             if (j <> h)
                             {
-                                l = insert(l,list(graph32(list(4,1,1),i,j,k,h),8),size(l));
-                                l = insert(l,list(graph32(list(3,2,1),i,j,k,h),6),size(l));
-                                l = insert(l,list(graph32(list(2,2,2),i,j,k,h),48),size(l));
+                                l[size(l)+1] = list(graph32(list(4,1,1),i,j,k,h),8);
+                                l[size(l)+1] = list(graph32(list(3,2,1),i,j,k,h),6);
+                                l[size(l)+1] = list(graph32(list(2,2,2),i,j,k,h),48);
                             }
                             for (m=0;m<=r;m++)
@@ -430 +435 @@
                                 if (k <> h and h <> m)
                                 {
-                                    l = insert(l,list(graph41(list(3,1,1,1),i,j,k,h,m),3),size(l));
-                                    l = insert(l,list(graph41(list(1,3,1,1),i,j,k,h,m),3),size(l));
-                                    l = insert(l,list(graph41(list(2,2,1,1),i,j,k,h,m),4),size(l));
-                                    l = insert(l,list(graph41(list(2,1,2,1),i,j,k,h,m),4),size(l));
-                                    l = insert(l,list(graph41(list(2,1,1,2),i,j,k,h,m),8),size(l));
-                                    l = insert(l,list(graph41(list(1,2,2,1),i,j,k,h,m),8),size(l));
+                                    l[size(l)+1] = list(graph41(list(3,1,1,1),i,j,k,h,m),3);
+                                    l[size(l)+1] = list(graph41(list(1,3,1,1),i,j,k,h,m),3);
+                                    l[size(l)+1] = list(graph41(list(2,2,1,1),i,j,k,h,m),4);
+                                    l[size(l)+1] = list(graph41(list(2,1,2,1),i,j,k,h,m),4);
+                                    l[size(l)+1] = list(graph41(list(2,1,1,2),i,j,k,h,m),8);
+                                    l[size(l)+1] = list(graph41(list(1,2,2,1),i,j,k,h,m),8);
                                 }
                                 if (k <> h and k <> m)
                                 {
-                                    l = insert(l,list(graph42(list(3,1,1,1),i,j,k,h,m),6),size(l));
-                                    l = insert(l,list(graph42(list(1,3,1,1),i,j,k,h,m),6),size(l));
-                                    l = insert(l,list(graph42(list(1,1,3,1),i,j,k,h,m),3),size(l));
-                                    l = insert(l,list(graph42(list(2,2,1,1),i,j,k,h,m),8),size(l));
-                                    l = insert(l,list(graph42(list(1,1,2,2),i,j,k,h,m),8),size(l));
-                                    l = insert(l,list(graph42(list(2,1,2,1),i,j,k,h,m),4),size(l));
-                                    l = insert(l,list(graph42(list(1,2,2,1),i,j,k,h,m),4),size(l));
+                                    l[size(l)+1] = list(graph42(list(3,1,1,1),i,j,k,h,m),6);
+                                    l[size(l)+1] = list(graph42(list(1,3,1,1),i,j,k,h,m),6);
+                                    l[size(l)+1] = list(graph42(list(1,1,3,1),i,j,k,h,m),3);
+                                    l[size(l)+1] = list(graph42(list(2,2,1,1),i,j,k,h,m),8);
+                                    l[size(l)+1] = list(graph42(list(1,1,2,2),i,j,k,h,m),8);
+                                    l[size(l)+1] = list(graph42(list(2,1,2,1),i,j,k,h,m),4);
+                                    l[size(l)+1] = list(graph42(list(1,2,2,1),i,j,k,h,m),4);
                                 }
                                 if (j <> h and j <> m)
                                 {
-                                    l = insert(l,list(graph43(list(3,1,1,1),i,j,k,h,m),18),size(l));
-                                    l = insert(l,list(graph43(list(2,2,1,1),i,j,k,h,m),16),size(l));
+                                    l[size(l)+1] = list(graph43(list(3,1,1,1),i,j,k,h,m),18);
+                                    l[size(l)+1] = list(graph43(list(2,2,1,1),i,j,k,h,m),16);
                                 }
                                 for (n=0;n<=r;n++)
@@ -456 +461 @@
                                     if (k <> h and h <> m and m <> n)
                                     {
-                                        l = insert(l,list(graph51(list(2,1,1,1,1),i,j,k,h,m,n),2),size(l));
-                                        l = insert(l,list(graph51(list(1,2,1,1,1),i,j,k,h,m,n),2),size(l));
-                                        l = insert(l,list(graph51(list(1,1,2,1,1),i,j,k,h,m,n),4),size(l));
+                                        l[size(l)+1] = list(graph51(list(2,1,1,1,1),i,j,k,h,m,n),2);
+                                        l[size(l)+1] = list(graph51(list(1,2,1,1,1),i,j,k,h,m,n),2);
+                                        l[size(l)+1] = list(graph51(list(1,1,2,1,1),i,j,k,h,m,n),4);
                                     }
                                     if (k <> h and h <> m and h <> n)
                                     {
-                                        l = insert(l,list(graph52(list(2,1,1,1,1),i,j,k,h,m,n),4),size(l));
-                                        l = insert(l,list(graph52(list(1,2,1,1,1),i,j,k,h,m,n),4),size(l));
-                                        l = insert(l,list(graph52(list(1,1,2,1,1),i,j,k,h,m,n),4),size(l));
-                                        l = insert(l,list(graph52(list(1,1,1,2,1),i,j,k,h,m,n),2),size(l));
+                                        l[size(l)+1] = list(graph52(list(2,1,1,1,1),i,j,k,h,m,n),4);
+                                        l[size(l)+1] = list(graph52(list(1,2,1,1,1),i,j,k,h,m,n),4);
+                                        l[size(l)+1] = list(graph52(list(1,1,2,1,1),i,j,k,h,m,n),4);
+                                        l[size(l)+1] = list(graph52(list(1,1,1,2,1),i,j,k,h,m,n),2);
                                     }
                                     if (k <> h and k <> m and k <> n)
                                     {
-                                        l = insert(l,list(graph53(list(2,1,1,1,1),i,j,k,h,m,n),12),size(l));
-                                        l = insert(l,list(graph53(list(1,2,1,1,1),i,j,k,h,m,n),12),size(l));
-                                        l = insert(l,list(graph53(list(1,1,2,1,1),i,j,k,h,m,n),4),size(l));
+                                        l[size(l)+1] = list(graph53(list(2,1,1,1,1),i,j,k,h,m,n),12);
+                                        l[size(l)+1] = list(graph53(list(1,2,1,1,1),i,j,k,h,m,n),12);
+                                        l[size(l)+1] = list(graph53(list(1,1,2,1,1),i,j,k,h,m,n),4);
                                     }
                                     if (j <> h and h <> m and h <> n)
                                     {
-                                        l = insert(l,list(graph54(list(2,1,1,1,1),i,j,k,h,m,n),4),size(l));
-                                        l = insert(l,list(graph54(list(1,1,2,1,1),i,j,k,h,m,n),16),size(l));
+                                        l[size(l)+1] = list(graph54(list(2,1,1,1,1),i,j,k,h,m,n),4);
+                                        l[size(l)+1] = list(graph54(list(1,1,2,1,1),i,j,k,h,m,n),16);
                                     }
                                     if (k <> h and k <> m and h <> n)
                                     {
-                                        l = insert(l,list(graph55(list(2,1,1,1,1),i,j,k,h,m,n),2),size(l));
-                                        l = insert(l,list(graph55(list(1,2,1,1,1),i,j,k,h,m,n),2),size(l));
-                                        l = insert(l,list(graph55(list(1,1,1,2,1),i,j,k,h,m,n),4),size(l));
+                                        l[size(l)+1] = list(graph55(list(2,1,1,1,1),i,j,k,h,m,n),2);
+                                        l[size(l)+1] = list(graph55(list(1,2,1,1,1),i,j,k,h,m,n),2);
+                                        l[size(l)+1] = list(graph55(list(1,1,1,2,1),i,j,k,h,m,n),4);
                                     }
                                     if (j <> h and j <> m and j <> n)
                                     {
-                                        l = insert(l,list(graph56(list(2,1,1,1,1),i,j,k,h,m,n),48),size(l));
+                                        l[size(l)+1] = list(graph56(list(2,1,1,1,1),i,j,k,h,m,n),48);
                                     }
                                     for (p=0;p<=r;p++)
@@ -492 +497 @@
                                         if (k <> h and h <> m and m <> n and n <> p)
                                         {
-                                            l = insert(l,list(graph61(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2),size(l));
+                                            l[size(l)+1] = list(graph61(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2);
                                         }
                                         if (k <> h and h <> m and m <> n and m <> p)
                                         {
-                                            l = insert(l,list(graph62(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2),size(l));
+                                            l[size(l)+1] = list(graph62(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2);
                                         }
                                         if (k <> h and h <> m and h <> n and n <> p)
                                         {
-                                            l = insert(l,list(graph63(list(1,1,1,1,1,1),i,j,k,h,m,n,p),1),size(l));
+                                            l[size(l)+1] = list(graph63(list(1,1,1,1,1,1),i,j,k,h,m,n,p),1);
                                         }
                                         if (k <> h and h <> m and h <> n and h <> p)
                                         {
-                                            l = insert(l,list(graph64(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6),size(l));
+                                            l[size(l)+1] = list(graph64(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6);
                                         }
                                         if (k <> h and k <> m and k <> n and n <> p)
                                         {
-                                            l = insert(l,list(graph65(list(1,1,1,1,1,1),i,j,k,h,m,n,p),4),size(l));
+                                            l[size(l)+1] = list(graph65(list(1,1,1,1,1,1),i,j,k,h,m,n,p),4);
                                         }
                                         if (k <> h and k <> m and m <> p and h <> n)
                                         {
-                                            l = insert(l,list(graph66(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6),size(l));
+                                            l[size(l)+1] = list(graph66(list(1,1,1,1,1,1),i,j,k,h,m,n,p),6);
                                         }
                                         if (j <> h and h <> m and m <> n and m <> p)
                                         {
-                                            l = insert(l,list(graph67(list(1,1,1,1,1,1),i,j,k,h,m,n,p),8),size(l));
+                                            l[size(l)+1] = list(graph67(list(1,1,1,1,1,1),i,j,k,h,m,n,p),8);
                                         }
                                         if (j <> h and h <> m and h <> n and h <> p)
                                         {
-                                            l = insert(l,list(graph68(list(1,1,1,1,1,1),i,j,k,h,m,n,p),12),size(l));
+                                            l[size(l)+1] = list(graph68(list(1,1,1,1,1,1),i,j,k,h,m,n,p),12);
                                         }
                                         if (j <> h and h <> m and h <> n and n <> p)
                                         {
-                                            l = insert(l,list(graph69(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2),size(l));
+                                            l[size(l)+1] = list(graph69(list(1,1,1,1,1,1),i,j,k,h,m,n,p),2);
                                         }
                                         if (k <> h and k <> m and k <> n and k <> p)
                                         {
-                                            l = insert(l,list(graph610(list(1,1,1,1,1,1),i,j,k,h,m,n,p),24),size(l));
+                                            l[size(l)+1] = list(graph610(list(1,1,1,1,1,1),i,j,k,h,m,n,p),24);
                                         }
                                         if (j <> h and j <> m and j <> n and j <> p)
                                         {
-                                            l = insert(l,list(graph611(list(1,1,1,1,1,1),i,j,k,h,m,n,p),720),size(l));
+                                            l[size(l)+1] = list(graph611(list(1,1,1,1,1,1),i,j,k,h,m,n,p),720);
                                         }
                                     }
@@ -548 +553 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0,(x),dp;
+    ring r = 0,x,dp;
     variety P = projectiveSpace(4);
     stack M = moduliSpace(P,2);
@@ -573 +578 @@
     for (i=0;i<=n;i++)
     {
-        l = insert(l,number(5^i),size(l));
+        l = insert(l,number(10^i),size(l));
     }
     return (l);
@@ -580 +585 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0,(x),dp;
+    ring r = 0,x,dp;
     variety P = projectiveSpace(4);
     def L = torusList(P);
@@ -657 +662 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0,(x),dp;
+    ring r = 0,x,dp;
     variety P = projectiveSpace(4);
     stack M = moduliSpace(P,2);
@@ -691 +696 @@
         int d = e[j][3];
         number c = (-1)^d*factorial(d)^2;
-        number y = c*(L[e[j][1]+1]-L[e[j][2]+1])^(2*d)/(d^(2*d));
+        number y = c*(L[e[j][1]+1]-L[e[j][2]+1])^(2*d)/(number(d)^(2*d));
         for (k=0;k<=n;k++)
         {
@@ -749 +754 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0,(x),dp;
+    ring r = 0,x,dp;
     variety P = projectiveSpace(4);
     stack M = moduliSpace(P,2);
@@ -788 +793 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0,(x),dp;
+    ring r = 0,x,dp;
     multipleCover(1);
     multipleCover(2);
@@ -794 +799 @@
     multipleCover(4);
     multipleCover(5);
+    multipleCover(6);
 }
 
@@ -812 +818 @@
     def F = fixedPoints(M);
     int i;
-    poly r = 0;
+    number r = 0;
     for (i=1;i<=size(F);i++)
     {
@@ -826 +832 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0,(x),dp;
+    ring r = 0,x,dp;
     linesHypersurface(2);
     linesHypersurface(3);
@@ -874 +880 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0,(x),dp;
+    ring r = 0,x,dp;
     rationalCurve(1);
     /*
@@ -880 +886 @@
     rationalCurve(3);
     rationalCurve(4);
-    rationalCurve(5);
     rationalCurve(1,list(4,2));
     rationalCurve(1,list(3,3));
@@ -897 +902 @@
     rationalCurve(4,list(3,2,2));
     rationalCurve(4,list(2,2,2,2));
-    rationalCurve(5,list(4,2));
-    rationalCurve(5,list(3,3));
-    rationalCurve(5,list(3,2,2));
-    rationalCurve(5,list(2,2,2,2));
-    */
 }
 
@@ -921 +921 @@
 {
     "EXAMPLE:"; echo=2;
-    ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp;
+    ring r = 0,x,dp;
     graph G = makeGraph(list(list(0,1,list(0,1,2)),list(1,1,list(1,0,2))),
     list(list(0,1,2)));
@@ -943 +943 @@
 {
     "EXAMPLE:"; echo=2;
-    ring R=(0,x1,x2,x3,x4),(q1,q2,q3,q4,q5,q6),dp;
+    ring r = 0,x,dp;
     graph G = makeGraph(list(list(0,1,list(0,1,2)),list(1,1,list(1,0,2))),
     list(list(0,1,2)));
@@ -1373 +1373 @@
     list v3 = k,3,f4,f5,f7;
     list v4 = h,2,f6,f9;
-    list v5 = m,1,f8,f11;
+    list v5 = m,2,f8,f11;
     list v6 = n,1,f10;
     list v7 = p,1,f12;
@@ -1516 +1516 @@
 }
 
-////////////////////////////////////////////////////////////////////////////////
-/// Auxilary Static Procedures in this Library /////////////////////////////////
-////////////////////////////////////////////////////////////////////////////////
-
 proc part(poly f, int n)
 "USAGE:     part(f,n); f poly, n int
@@ -1646 +1642 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0,(x),dp;
+    ring r = 0,x,dp;
     poly f = 3+x;
     expp(f,3);
@@ -1718 +1714 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0, (h,e), wp(1,1);
+    ring r = 0,(h,e),wp(1,1);
     ideal rels = he,h2+e2;
     variety V = makeVariety(2,rels);
@@ -1744 +1740 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0, (h,e), wp(1,1);
+    ring r = 0,(h,e),wp(1,1);
     ideal rels = he,h2+e2;
     variety V = makeVariety(2,rels);
@@ -1769 +1765 @@
 {
     "EXAMPLE:"; echo=2;
-    ring r = 0, (h,e), wp(1,1);
+    ring r = 0,(h,e),wp(1,1);
     ideal rels = he,h2+e2;
     int d = 2;
@@ -2086 +2082 @@
 
 ////////////////////////////////////////////////////////////////////////////////
-////////// Procedures concerned with sheaves ///////////////////////////////////
+////////// Procedures concerned with abstract sheaves ///////////////////////////////////
 ////////////////////////////////////////////////////////////////////////////////