You have downloaded Stephen's computer search for exceptional collections on surfaces of general type with p_g=0. Congratulations! These scripts require Magma, and some of them work in the online calculator http://magma.maths.usyd.edu.au/calc/ To use these scripts, first load the main program <exceptional> in magma by typing: > load "exceptional"; This loads the main procedures and functions which are used in the search. There are separate files containing the data associated to each surface. For example, to study the Kulikov surface, type: > load "kulikov"; It may be better to load the data files interactively: > iload "kulikov"; Some surfaces take much longer to load than others. The possible attributes of a surface are listed by typing > Surface; But you should not do this, since the output would be a very long list! It is better to query a single attribute of the surface X, using "X`attribute", e.g. > X`Name; returns the name of the surface X. //////////////////////////////////////////////////// The following functions are probably the most useful, the others are mostly used behind the scenes. pushViaBasis(X,L,tau) -- given a multidegree L and a torsion twist tau, return the list of line bundles on Y whose direct sum is the pushforward of L(tau) e.g. > pushViaBasis(X,[1,-1,-1,0],T![1,0,0]); findAcyclic(X,L) -- given a multidegree L, return the list of torsion line bundles tau such that L(tau) is acyclic e.g. > findAcyclic(X,[1,-1,-1,-1]); findExceptional(surface,LL) -- given a surface and a sequence of multidegrees LL=[L1,...Ln], return the list of sequences of torsion twists [t1,...,tn] such that [Oh,L1(t1),...,Ln(tn)] is an exceptional collection on X (NB: we assume that the first entry in the exceptional collection is the structure sheaf Oh) e.g. > LL:=[[-1,0,0,0],[0,-1,0,0],[0,0,-1,0],[0,0,0,-1],[0,0,0,-2]]; > findExceptional(X,LL); findHoms(surface,LL,twists) -- given a surface, a sequence of multidegrees LL=[L1,...,Ln] and a sequence of torsion twists=[t1,...,tn], return two arrays. The (ij)th entry of the first array is the Euler--Poincare characteristic chi(E_i,E_{i+j}), where E_i=Li(ti). The (ij)th entry of the second array is the two pushforwards of E_{i+j}*E_i^{-1} and omega_X*E_i*E_{i+j}^{-1}. The two arrays together contain the necessary information for computing all Ext-groups of [Oh,L1(t1),...,Ln(tn)] e.g. > E,H:=findHoms(X,LL,twists); > E[[1,2]]; // returns chi(E_1,E_3) > H[[1,2]]; // returns pushforwards of E_3*E_1^{-1} and omega_X*E_1*E_3^{-1} Weylreflection(surface,L,alpha) -- given a line bundle L and a root alpha in Pic Y, returns the image of L under the Weyl reflection corresponding to alpha ////////////END////////////