Example 2. Modular curve of level 67
Consider the homogeneous polynomials

f := 5*v*z - 2*w*x - 3*w*y + w*z,

g := 10*v^2 + 5*w*v - 5*w^2 + 4*x^2 - 12*x*y + 2*x*y - 2*y^2 - 35*y*z - 12*z^2,

h := 15*v^2 - 5*w*v + 5*w^2 + 8*x^2 - 12*x*y - 14*x*z - 11*y^2 - 3*y*z + 15*z^2.

Then C := V(f,g,h) in P^4=Proj(K[x,y,z,v,w]) is a non-singular curve called the (classical) modular curve of level 67.

Assume char(K)=p=3.
By executing our code (text file) over Magma, we obtain a matrix representing the Frobenius F^* on H^1(C,O_C), and its rank is 3 (log file is here).

Example 3. Modular curve of level 23
Consider the homogeneous polynomials

f1 := y^2 + (- x3 - x1 - x0)*y + 2*x3*x2 + 3*x1^2 - 2*x1*x0 + 2*x0^2,

f2 := x1^2 - x0*x2,

f3 := x2^2 - x1*x3,

f4 := x3*x0 - x2*x1.

Then C:=V(f1,f2,f3,f4) in P^4=Proj(K[x0,x1,x2,x3,y]) is the normalization of the modular curve of level 23, which is a hyperelliptic curve of genus 2.

Assume char(K)=p=5.
By executing our code (text file) over Magma, we obtain a matrix representing the Frobenius F^* on H^1(C,O_C), and its rank is 2 (log file is here).

Example 4. Horrocks-Mumford surfaces

A HM surface is embedded into the 4 projective space as an intersection of 3 quintics and 15 sextics.

Each of the following text files includes a set of defining polynomials for a HM surface.

• HM-surface example1.txt: An example of a HM surface X for which the rank of the Frobenius F^* on H^1(X,O_X) is 2.

• HM-surface example2.txt: An example of a HM surface X for which the rank of the Frobenius F^* on H^1(X,O_X) is 1.

• HM-surface example3.txt: An example of a HM surface X for which the rank of the Frobenius F^* on H^1(X,O_X) is 0.

Here is our Magma code (main code), which computes a matrix representing F^* for the first example.

Before loading this main code, please also

1. Put these text files (file 1, file 2) in an appropriate directly. Assume that the name of the directly is ``C:/user" here.

2. Edit the main code as follows: Correct the sentences load"C:/Users/m-kudo/Desktop/Basis.txt" and load"C:/Users/m-kudo/Desktop/Frob1.txt" to load"C:/user/Basis.txt" and load"C:/user/Frob1.txt" respectively.

Our log file obtained by the main code is here.

Remark. Note that a basis of H^1 and a representation matrix Y for F^* obtained by the execution can be different from those written in our paper, since Magma's built-in function "FreeResolution" outputs minimal free resolutions (for the same coordinate ring S/I) which can be different for each execution. More specifically, two minimal free resolutions F_* and F'_* outputted by FreeResolution for S/I are definitely isomorphic to each other, but there can exist i such that representation matrices of the i-th differential maps F_i -> F_{i-1} and F'_{i} -> F'_{i-1} are different.
(Of course, the rank of Y (and its characteristic polynomial) takes the same value even for different resolutions of the same X.)