FdotBjIntMat -- matrix of intersection numbers between F-curves and divisors on $\bar{M}_{0,n}$

Synopsis

Usage:

FdotBjIntMat(n)
Inputs:
- n, an integer,
Outputs:
- M, a matrix,

Description

This function produces the matrix of intersection numbers between the standard basis of $S_n$ symmetric divisors and the most popular basis of $S_n$ symmetric F curves. Specifically, the i,j-th entry of the matrix is $F_{n-i-2,i,1,1} . B_j$. This matrix can be used for instance to write a divisor in the standard basis if its intersection numbers with the F curves are known. See [AGSS] Section 4 for explicit formulas.

These intersection numbers are integers, but we create the matrix over the rational numbers so that Macaulay2 will invert it correctly if we want to do so later.

In the example below, we use this function to find the divisor class of an $S_{12}$ symmetric divisor $D$ on $\bar{M}_{0,12}$ such that $D . F_{1,1,i,12-i-2} = 1$ if $i=0$, and 0 otherwise. Then we check that $D$ has the correct intersection numbers.

i1 : M=FdotBjIntMat(12)

o1 = | 3 -1 0  0  0  |
     | 0 2  -1 0  0  |
     | 1 -1 2  -1 0  |
     | 1 0  -1 2  -1 |
     | 1 0  0  -2 2  |

              5       5
o1 : Matrix QQ  <-- QQ

i2 : N=M^-1

o2 = | 2/11   2/11  2/11  2/11  1/11  |
     | -5/11  6/11  6/11  6/11  3/11  |
     | -10/11 1/11  12/11 12/11 6/11  |
     | -13/11 -2/11 9/11  20/11 10/11 |
     | -14/11 -3/11 8/11  19/11 15/11 |

              5       5
o2 : Matrix QQ  <-- QQ

i3 : v=N*(matrix{{1},{0},{0},{0},{0}})

o3 = | 2/11   |
     | -5/11  |
     | -10/11 |
     | -13/11 |
     | -14/11 |

              5       1
o3 : Matrix QQ  <-- QQ

i4 : D=symmetricDivisorM0nbar(12,flatten entries v)

      2       5      10      13      14
o4 = --*B  - --*B  - --*B  - --*B  - --*B
     11  2   11  3   11  4   11  5   11  6

o4 : S_12-symmetric divisor on M-0-12-bar

i5 : symmetricCurveDotDivisorM0nbar({1,1,1,9},D)

o5 = 1

o5 : QQ

i6 : apply(5, i-> symmetricCurveDotDivisorM0nbar({1,1,i+1,12-i-3},D))

o6 = {1, 0, 0, 0, 0}

o6 : List

Ways to use FdotBjIntMat:

FdotBjIntMat(ZZ)

For the programmer

The object FdotBjIntMat is a method function.