Around Aug 2005. This is the first appearance of Porb in a general context. It contains experiments with curves orbifold strata on surfaces and 3-fold, etc. that are not CY, typically wPP^2 such as PP(a,b,c). The "disaster" possibly refers to curve cases when the 1/[1,1,r,r] principal part is more than just b*Qorb(1/r(a,r-a),r)/(1-t^r). Possibly the best statement for curve of type 1/3(1) is: its effect in k = 6 al-1 is t^(3 al+3)/(1-t)(1-t^3)^2 in k = 6 al-4 is -t^(3 al+1) - t^(3 al+2)/(1-t)(1-t^3)^2. That is wrong. Curve of type 1/3(1). Even for fixed k, the degree of the normal bundle varies. S = PP(1,3,3); the whole of P is from the degree term of Ga: > 1/denom([1,3,3]) eq Porb(3,[1],-4)/(1-t^3); // true S4 in PP(1,1,3,3) has k = -4, N = Oh(2) Since PI = 0, the whole P is from the curve, so PC(1/3(1),deg = 1/9, N=Oh(2) in k = -4) = (1+t+t^2+t^3)/(1-t)(1-t^3)^2. > d:=4; L:= [1,1,3,3]; (1-t^d)/denom(L) - Porb(3,[1],-1)/(1-t^3); (-t - 1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) (too simple to give diagnostic data; this could agree with > Porb(3,[2],-5)*t/(1-t); // mult by t changes Gor deg (-t - 1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) or with > Porb(3,[1],-4)*(1+t)/(1-t); (-t - 1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) or something else. S5 in PP(1,2,3,3) has k = -4, N = Oh(-1), subtract Porb(1/2(1,1), k = -4) to get PC(1/3(1), deg = 1/9, N = Oh(-1) in k = -4) = (-t-t^2)/(1-t)(1-t^3)^2. > d:=5;L:=[1,2,3,3];(1-t^d)/denom(L)-Porb(2,[3,3],-4)-Porb(3,[1],-1)/(1-t^3); 0 // that is, no N terms. This one has accidental node. S7 in PP(1,4,3,3) k = -4, N = Oh(+2), subtract Porb(1/4(1,3), k = -4) to get PC(1/3(1), deg = 1/9, N = Oh(+2) in k = -4) = (1+t+t^2+t^3)/(1-t)(1-t^3)^2. S8 in PP(1,5,3,3) has k = -4, N = Oh(-1), subtract Porb(1/5(1,3), k = -4) to get PC(1/3(1), deg = 1/9, N = Oh(-1) in k = -4) = (-t-t^2)/(1-t)(1-t^3)^2. for d == 1 mod 3 Sd in PP(1,(d-3),3,3) has k = -4, N = Oh(+2), need to subtract Porb(1/(d-3)(1,3),-4) to get d:=13; (1-t^d)/denom([1,d-3,3,3]) - Porb(d-3,[1,3],-4); (-t^3 - t^2 - t - 1)/(t^7 - t^6 - 2*t^4 + 2*t^3 + t - 1) > for i in [1..5] do for> d:=3*i+4;L:= [1,3*i+1,3,3];(1-t^d)/denom(L)-Porb(3*i+1,[1,3],-4)-Porb(3,[1],-1)/(1-t^3); for> end for; (-t - 1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) // in each case for d == 2 mod 3 Sd in PP(1,(d-3),3,3) has k = -4, N = Oh(-1), need to subtract Porb(1/(d-3)(1,3),-4) to get d:=14; (1-t^d)/denom([1,d-3,3,3]) - Porb(d-3,[1,3],-4); (t^2 + t)/(t^7 - t^6 - 2*t^4 + 2*t^3 + t - 1) > for i in [11..15] do for> d:=3*i+5;L:= [1,3*i+2,3,3];(1-t^d)/denom(L)-Porb(3*i+2,[1,3],-4)-Porb(3,[1],-1)/(1-t^3); for> end for; 0 // in each case That's nice and convincing. S6 in P(1,3,3,3). This a curve of deg 2/9 with N = Oh(-1); PC(1/3(1),deg = 2/9, N=Oh(-1) in k = -4) = (1+t^3)/(1-t)(1-t^3)^2. > d:=6;L:= [1,3,3,3];(1-t^d)/denom(L)-2*Porb(3,[1],-1)/(1-t^3); (-t - 1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) S12 in PP(4,6,3,3) has k = -4, one 1/2 point on first PP^1, N = Oh(-4) PC(1/3(1), deg = 2/9, N = Oh(-4) in k = -4) = (-2t-2t^2)/(1-t)(1-t^3)^2. > d:=12;L:= [4,6,3,3];(1-t^d)/denom(L)-Porb(2,[3,3],-4)-2*Porb(3,[1],-1)/(1-t^3); 0 S18 in PP(4,6,3,9) has k = -4, one 1/4(1,3), one 1/2, N = Oh(-4), gives > d:=18;L:= [4,6,3,9];(1-t^d)/denom(L)-Porb(2,[3,3],-4) -Porb(4,[1,3],-4)-Porb(3,[1],-1)/(1-t^3); 0 (1-t^30)/denom([10,3,6,15]) - Porb(5,[3,1],-4) - Porb(2,[1,1],-4); > d:=30;L:= [10,3,6,15];(1-t^d)/denom(L)-Porb(2,[1,1],-4)-Porb(5,[3,6],-4)-Por\ b(3,[1],-1)/(1-t^3); (t + 1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) // that is, minus the N term P(4,3,3). Curve of deg 1/9 with N = Oh(-4) in k = -10; need to subtract Porb(1/4(3,3),-10) to get 1/denom([4,3,3]) - Porb(4,[3,3],-10); (t + 1)/(t^9 - t^8 - 2*t^6 + 2*t^5 + t^3 - t^2) that is PC(1/3(1), deg = 1/9, N = Oh(-4) in k = -10) = (-t^-2 - t^-1)/(1-t)(1-t^3)^2. > 1/denom([4,3,3])-Porb(4,[3,3],-10)-Porb(3,[1],-7)/(1-t^3); 0 Now try all the same things with k = -1. S7 in PP(1,1,3,3) has k = -1, N = Oh(5) PI := (1-t+t^2)/(1-t)^3; (1-t^7)/denom([1,1,3,3]) - (1-t+t^2)/(1-t)^3; // (-t^2-t^4)/[1,3,3] > d:=7;L:= [1,1,3,3];(1-t^d)/denom(L)-PI-Porb(3,[1],2)/(1-t^3); t^2/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) this quantity is just a k-changed version of N-term: > t^2/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) > + (t+1)/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1); -1/(1-t^3) That is, one is t^2/denom([1,1,3], the other is (1+t)/ditto, so sum is divisible by (1-t^3) div (1-t) S8 in PP(1,2,3,3) has k = -1, N = Oh(-1); (1-t^8)/denom([1,2,3,3]) - (1-2*t+t^2)/(1-t)^3; // (t^2 + 2t^3 + t^4)/[1,3,3] > d:=8;L:= [1,2,3,3];(1-t^d)/denom(L)-(1-2*t+t^2)/(1-t)^3-Porb(3,[1],2)/(1-t^3); -t^2/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) S11 in PP(1,5,3,3) has k = -1, N = Oh(-1); 1/5(3,3) (1-t^11)/denom([1,5,3,3]) - (1-2*t+t^2)/(1-t)^3 - Porb(5,[3,3],-1); // (t^2 + 2*t^3 + t^4)/[1,3,3] > d:=11;L:=[1,5,3,3];(1-t^d)/denom(L)-(1-2*t+t^2)/(1-t)^3-Porb(5,[3,3],-1)-Por\ b(3,[1],2)/(1-t^3); -t^2/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) S14 in PP(2,7,3,3) has k = -1, N = Oh(-7); (1-t^14)/denom([2,7,3,3]) - (1-3*t+t^2)/(1-t)^3; // (3*t^2 + 4*t^3 + 3*t^4)/[1,3,3] > d:=14;L:=[2,7,3,3];(1-t^d)/denom(L)-(1-3*t+t^2)/(1-t)^3-Porb(3,[1],2)/(1-t^3); -3*t^2/(t^5 - 2*t^4 + t^3 - t^2 + 2*t - 1) a:=-2; b:=0; c:=1; X:=(1+a*t+t^2)/(1-t)^3 + b*t^2/denom([1,1,3]) + c*t^3/&*[1-t^i : i in [1,3,3]]; XX := X*(1-t)*(1-t^3)^2; XX; This works in many cases, with P1 = 3+a, c = deg Ga, and b either decreases genus of Ga by 1 or add a disjoint isolated 1/3(2,2) point. Now try k = 2, c = 5, PI = (1+a*t+b*t^2+b*t^3+a*t^4+t^5)/(1-t)^3; S10 in PP(1,1,3,3) has PI = (1-t-t^4+t^5)/(1-t)^3; a:=-1;b:=0;c:=2;d:=1; PI:= (1+a*t+b*t^2+b*t^3+a*t^4+t^5)/(1-t)^3; X:= PI + c*t^3*(1+t)/denom([1,1,3]) - d*t^4*(1+t)/&*[1-t^i : i in [1,3,3]]; XX:=X*&*[1-t^i:i in [1,3,3]]; XX; I think the statement for a pure m-dim orbifold locus of transverse type 1/r[L] in the context of K = kA is P = P_Initial + P_Orb where P_Orb is a sum of P_i / (1-t)^{i+1} * (1-t^r)^{m-i} and the P_i are Laurent polynomials, the same as in the 0-dim orbifold formula shifted to a suitable interval to give right symmetric degree. Surface S13 in PP(1,1,4,4) has curve Ga of 1/4(1) and k = 3 so c = 6. > PI := (1-t-t^5+t^6)/(1-t)^3; > A := (t^4+t^5)/denom([1,1,4]); > B := (-t^5-t^6-t^7)/denom([1,4,4]); > (PI+2*A+B)*denom([1,1,4,4]); -t^13 + 1 > d:=13;L:=[1,1,4,4];(1-t^d)/denom(L)-(1-t-t^5+t^6)/(1-t)^3-Porb(4,[1],7)/(1-t^4); -2*t^4/(t^5 - 3*t^4 + 4*t^3 - 4*t^2 + 3*t - 1) > d:=16;L:=[1,4,4,4];(1-t^d)/denom(L)-(1-2*t+t^2+t^4-2*t^5+t^6)/(1-t)^3 -4*Porb(4,[1],7)/(1-t^4); -2*t^4/(t^5 - 3*t^4 + 4*t^3 - 4*t^2 + 3*t - 1) You can try out PI+a*A+b*B in a range of values of a,b; e.g., > (PI+A+B)*denom([1,1,4,8,12]); t^29 - t^24 - t^5 + 1 Predicts S(5,24) in PP(1,1,4,8,12). Note that genus Ga = 1 > (PI+3*A+B)*denom([1,1,4,4,4,5]); -t^22 + 2*t^14 + 2*t^13 + t^12 - t^10 - 2*t^9 - 2*t^8 + 1 A Pfaffian 3 3 4 4 \\ 4 5 5 \\ 5 5 \\ 6. I think that Ga is PP(z1,z2) and PP(z3) is an isolated nonquasismooth point. Maybe a mirage. > (PI+3*A+2*B)*denom([1,1,4,4,4]); t^17 - t^9 - t^8 + 1 Predicts S(8,9) in PP(1,1,4,4,4) > (PI+3*A+3*B)*denom([1,1,4,4,4]); t^17 - t^12 - t^5 + 1 Predicts S(5,12) in PP(1,1,4,4,4) Here b = deg Ga the orbifold curve 1/4(1), whereas increasing a (?) possibly increments genus(Ga) or adds isolated or embedded points In the same way, B is easy to understand as Porb(4,[1],6)/(1-t^4), whereas A with its factor of (1+t) is more subtle. There is also a hint that B only occurs in the comb A+B = (t^4+t^5+t^6+t^7+t^8)/ denom([1,4,4]. --------------------- Let's try some 5s. PP(1,5,5) has k=-11; Porb(1/5(1),-1) = -(t+t^2+t^3+t^4)/[1,5] A := Porb(5,[1],-1)/(1-t^5); B := (1+t)/denom([1,1,5]); A+B = (1-t^6)/[1,1,5,5] 2A+B = (1-t^10)/[1,5,5,5] 3A+B has P2 < 0; Not much pleasure to get from this. Let's do k = -1. PI := (1-t+t^2)/denom([1,1,5,5]); A := Porb(5,[1],4)/(1-t^5); // i.e. t^5/[1,5,5] B := (-t^2-t^3-t^4)/denom([1,1,5]); > (PI+A+B)*denom([1,1,5,5]); -t^11 + 1 PI+A+2B has P2 = 2, so would have relation x1x2=0. > (PI+2*A+B)*denom([1,1,5,5,5]); t^16 - t^10 - t^6 + 1 > (PI+3*A+B)*denom([1,1,5,5,5,5]); -t^21 + 2*t^15 + 3*t^11 - 3*t^10 - 2*t^6 + 1 Change to PI = (1-2t+t^2) to get > (PI+A+0*B)*denom([1,5,10,15]); -t^30 + 1 > (PI+2*A+0*B)*denom([1,5,5,10]); -t^20 + 1 > (PI+3*A)*denom([1,5,5,5]); -t^15 + 1 > (PI+4*A)*denom([1,5,5,5,5]); t^20 - 2*t^10 + 1 -------------------------- PP(2,5,5) has k=-12 and 1/2(1,1) A:=Porb(5,[2],-7)/(1-t^5); B:=(-1/t-1/t^4)/denom([1,1,5]); > A+B+Porb(2,[1,1],-12); -1/(t^12 - t^10 - 2*t^7 + 2*t^5 + t^2 - 1), i.e. 1/[2,5,5] do k=-2: PI := (1+t)/(1-t)^3; // no choice A:=Porb(5,[2],3)/(1-t^5); // i.e. (t^3+t^6)/[1,5,5] > B2:=(2*t+t^2+t^3+2*t^4)/denom([1,1,5]); > B3:=(t+t^2+t^3+t^4)/denom([1,1,5]); > B4:= (-t-t^4)/denom([1,1,5]); NB B3 = (1+t)/(1-t)*Porb(5,[2],-2); and B2 = 2B3 + B4; // B:=(-t-t^3-t^5)/denom([1,2,5]); // BB := (t+t^2+t^3+t^4)/denom([1,1,5]); S12 in PP(2,2,5,5) has k = -2 and 6 x 1/2(1,1) > (1-t^12)/denom([2,2,5,5]) - PI - 6*Porb(2,[1,1],-2)-A; (-2*t^4 - t^3 - t^2 - 2*t)/(t^7 - 2*t^6 + t^5 - t^2 + 2*t - 1) B:=(2*t+t^2+t^3+2*t^4)/denom([1,1,5]); Problem: I sometimes need to use this B, sometimes BB = (t+t^4)/[1,1,5]. which is right? They differ by (t+t^2+t^3+t^4)/[1,1,5] do k = +3, c = 6, PI := (1+a*t+b*t^2+c*t^3+b*t^4+c*t^5+t^6)/(1-t)^3; A:=Porb(5,[2],8)/(1-t^5); // i.e. (t^6+t^8)/[1,5,5] Result for a (pure) curve Ga of type 1/r(a1,..a{n-1}), k == -sum ai mod 5 ==================================== General formula is P0 + P1 + P2 P0 = Q0/(1-t)^{n+1}; here Q0 is a symmetric (genuine) polynomial of deg c = k+n+1 built out of the first [c/2] (round-down) plurigenera. If c < 0 then P0 = 0. P1 = Q1/(1-t)^n(1-t^r); here P1 is a symmetric Laurent polynomial of symmetric degree k+n+r with support in the interval k+n+r/2 .. k+n+3r/2 P1 is not allowed to change terms up to [c/2] = [(k+n+1)/2] P2 = d*Q2/(1-t)^{n-1}(1-t^r)^2 where Q2 = Qorb(r,[a1,..a{n-1}],k-r), a symmetric Laurent polynomial of symmetric degree k+n+r-1 coming from isolated sing 1/r[a1,..a{n-1}] in deg k For example, for a (pure) curve Ga of type 1/5(2), k = -2 so c = 1. Q0 = 1+t; Q1 = at+bt^2+bt^3+at^4 Qorb(5,[2],3) = -t^3-t^6 So P = P0 + P1 + P2 = function PP(a,b,d); P0 := (1+t)/(1-t)^3; // doesn't affect any plurigenera except p0 = 1 P1 := (a*t+b*t^2+b*t^3+a*t^4)/((1-t)^2*(1-t^5)); // affects p1, p2 P2 := d*(-t^3-t^6)/((1-t)*(1-t^5)^2); // affects p3 P := P0 + P1 + P2; return P*denom([1,5,5]); end function; Try to do CY 3folds with 1/r(1,r-1), k = 0 function PP(p1,p2,a,b,d) Q0 := 1 + p1*t + p2*t^2 + p1*t^3 + t^4; P0 := Q0/(1-t)^4; Q1 := a*t^3 + b*t^4 + a*t^5; P1 := Q1/((1-t)^3*(1-t^5)); P2 := d*Porb(5,[1,4],5)/(1-t^5); return P0+P1+P2; end function; > PP(-3,3,2,2,4) *denom([1,4,5,5,5]); -t^20 + 1 > PP(-2,1,2,2,1) *denom([1,1,4,5,5]); -t^16 + 1 function PP(p1,p2,a,b,d) Q0 := 1 + p1*t + p2*t^2 + p1*t^3 + t^4; P0 := Q0/(1-t)^4; Q1 := a*t^3 + b*t^4 + a*t^5; P1 := Q1/((1-t)^3*(1-t^5)); P2 := d*Porb(5,[1,4],5)/(1-t^5); return P0+P1+P2; end function; function PP(p1,p2,a,b,d) // this is for a curve of 1/5(1,4) Q0 := 1 + p1*t + p2*t^2 + p1*t^3 + t^4; P0 := Q0/(1-t)^4; Q1 := a*t^3 + b*t^4 + a*t^5; P1 := Q1/((1-t)^3*(1-t^5)); P2 := d*Porb(5,[2,3],5)/(1-t^5); return P0+P1+P2; end function; Rule: The contr from a pure curve Ga of 1/r(1,r-1) is 1/denom([1,1,r,r]) * ( -b*t^{r+1} + a*t^3*((1-t^(r-2)) div (1-t)) * ((1-t^r) div (1-t)) ); where b = deg Ga (in units of 1/r^2) and deg_Ga(E1-E{r-1}) = 2*r*a - 2*b. or better -b*t^{r+1} / denom([1,1,r,r]) + a*t^3*((1-t^(r-2)) div (1-t))) / denom([1,1,1,r]); The sum in the second numerator is t^3 + .. + t^r (with r-2 terms), which is t * (Scores((r-1)/r) - Scores(1/r)) //=========================== function Porb(r,L,k) if &or[GreatestCommonDivisor([r,a]) ne 1 : a in L] // other mugtrap then error "Error in Porb: Not Coprime"; end if; if (k + &+[i : i in L]) mod r ne 0 then error "Error in Porb: Canonical weight not compatible"; end if; A := (t^r-1) div (t-1); l:=Ceiling((k+#L+1)/2); de := Maximum(0,Ceiling(-l/r)); // this is a kludge to avoid programming Laurent polynomials properly m := l + de*r; B := t^m*&*[(1-t^i) div (1-t) : i in L]; h_throwaway, al_throwaway, be := XGCD(A,B); return t^m*be/((1-t)^#L*(1-t^r)*t^(de*r)); end function; //=========================== function PI(L) // only for CY 3folds n1 := #[i : i in L | i eq 1]; n2 := #[i : i in L | i eq 2]; return (1 + (n1-4)*t + (n2+Binomial(n1-3,2))*t^2 + (n1-4)*t^3 + t^4)/(1-t)^4; end function; //=========================== function PI(L,k) // assumes #L is dim n + 1 n := #L; //=========================== Phalf := -t^3/denom([1,1,2,2]); Qthird := t^3/denom([1,1,1,3]); // contributes t^3 Pthird := Porb(3,[1,2],3)/(1-t^3); // contributes -t^4 Q4th := (t^3+t^4)/denom([1,1,1,4]); P4th := Porb(4,[1,3],4)/(1-t^4); //=========================== function Scores(r,b) if GreatestCommonDivisor(r,b) ne 1 or b gt r// other mugtrap then error "Error in Scores: not coprime"; end if; return &+[t^i : i in [1..r] | (b*i div r) gt (b*(i-1) div r)]; end function; //=========================== // This function for the dissident points on CY 3folds doesn't work in general at present. // It assume (but doesn't check) that k = 0, #L = 3, sum ai == mod r, // and that every divisor of r divides at most one of the ai. /* if &or[GreatestCommonDivisor([r,a]) ne 1 : a in L] // other mugtrap then error "Error: Not Coprime"; end if; */ /* if (k + &+[i : i in L]) mod r ne 0 then error "Error: Canonical weight not compatible"; end if; */ de := Maximum(0,Ceiling(-l/r)); // this is a kludge to avoid programming Laurent polynomials properly OLD function Qorb(r,L,k) A := (t^r-1) div (t-1); B := denom(L) div(1-t)^#L; l := Degree(GCD(A,B))+3; // ?? probably + k + #L more generally h, al_throwaway, be := XGCD(A,t^l*B); return t^l*be/(h*denom([1,1,1,r])); end function; //=========================== function denom(L) return &*[1-t^i:i in L]; end function; m:=6;B:=t^m*&*[(1-t^i) div (1-t) : i in [1,2,3]]; h,al,be:= XGCD(A,B); t^m*be/(h*(1-t)^3*(1-t^6)); Y:=t^m*be/(h*(1-t)^3*(1-t^6)); d:=13;L:=[1,1,2,3,6];(1-t^d)/&*[1-t^i:i in L]-PI(L)-Y; Denominator($1) eq denom([1,1,1,3]); OLD function Qorb(r,L) // for a curve of transverse 1/r(L) in k = 0 l := Ceiling(r/2); A:=(t^r-1) div (t-1); B:=t^l*denom(L) div (1-t)^3; h,al,be:=XGCD(A,B); // be; return t^l*be/(h*denom([1,1,1,r])); end function; d:=13;L:=[1,1,3,3,5];(1-t^d)/denom(L)-PI(L)-Porb(5,[1,1,3],0) - Porb(3,[1,2],3)/(1-t^3); // 3*t^3/(t^6 - 3*t^5 + 3*t^4 - 2*t^3 + 3*t^2 - 3*t + 1) subtract one copy of Porb(3,[1,2],3)/(1-t^3) for every unit of deg Ga, and b*t^3/denom([1,1,1,3]) for some b, the N factor d:=19;L:=[1,1,3,5,9];(1-t^d)/denom(L)-PI(L)-Porb(5,[1,1,3],0)-Qorb(9,[1,3,5],0)-0*Porb(3,[1,2],3)/(1-t^3); // 2*t^3/(t^6 - 3*t^5 + 3*t^4 - 2*t^3 + 3*t^2 - 3*t + 1) > d:=16;L:=[1,1,3,3,8];(1-t^d)/denom(L)-PI(L)-0*Porb(5,[1,1,3],0)-Porb(3,[1,2],3)/(1-t^3); // 4*t^3/(t^6 - 3*t^5 + 3*t^4 - 2*t^3 + 3*t^2 - 3*t + 1) > d:=28;L:=[1,1,3,9,14];(1-t^d)/denom(L)-PI(L)-0*Porb(5,[1,1,3],0)-Qorb(9,[1,3,5],0)-0*Porb(3,[1,2],3)/(1-t^3); // 3*t^3/(t^6 - 3*t^5 + 3*t^4 - 2*t^3 + 3*t^2 - 3*t + 1) Experimental results: Y20 in PP(2,3,4,5,6) has 1/6(3,4,5), Porb of which is -t^6/denom(1,2,3,6) a curve of 1/2(1,1) of modified degree 2 a curve of 1/3(1,2) of modified degree 1 and N factor -t^3/denom(1,1,1,3) (1-t^d)/denom(L) eq PI(L) + Y + 2*Porb(2,[1,1],2)/(1-t^2) + Porb(3,[1,2],3)/(1-t^3) - t^3/denom([1,1,1,3]); "Take care of the plurigenera and the topology will take care of itself." > d:=9;L:=[1,1,1,2,4]; (1-t^d)/denom(L)-PI(L)-Qorb(4,[1,1,2],0); // 0 i.e. the 1/4(1,1,2) point eats 1/2 from deg Ga, leaving nothing > d:=10;L:=[1,1,2,2,4]; (1-t^d)/denom(L)-PI(L)-Qorb(4,[1,1,2],0)-2*Porb(2,[1,1],2)/(1-t^2); // 0 i.e. the 1/4(1,1,2) point eats 1/2 from deg Ga, leaving degree 2 > d:=18;L:=[1,1,2,6,8]; (1-t^d)/denom(L)-PI(L)-Qorb(8,[1,1,6],0); // 0 i.e. the 1/8(1,1,6) point eats 3/4 from deg Ga, leaving nothing > d:=24;L:=[1,1,6,8,8];(1-t^d)/denom(L)-PI(L)-3*Qorb(8,[1,1,6],0)+2*Porb(2,[1,1],2)/(1-t^2); // 0 i.e each of the 3 1/8(1,1,6) points eats 3/4 from deg Ga, leaving -2 > d:=20;L:=[1,1,2,6,10]; (1-t^d)/denom(L)-PI(L)-Qorb(6,[1,1,4],0); // 0 i.e. the 1/6(1,1,4) point eats 1/3 from deg Ga, leaving nothing > d:=21;L:=[1,1,2,7,10]; (1-t^d)/denom(L)-PI(L)-Qorb(10,[1,2,7],0); // 0 i.e. the 1/10(1,2,7) point eats 1/5 > d:=24;L:=[1,1,2,8,12]; (1-t^d)/denom(L)-PI(L)-Qorb(4,[1,1,2],0); // 0 d:=17;L:=[1,1,2,5,8];(1-t^d)/denom(L)-PI(L)-Porb(5,[1,1,3],0)-Qorb(8,[1,2,5],0)-Phalf; // 0 1/8(1,2,5) eats 5/4 x 1/4 of deg, leaving -1 x 1/4 > d:=48;L:=[2,3,4,16,23]; (1-t^d)/denom(L)-PI(L)-Porb(23,[3,4,16],0)-3*Qorb(4,[2,3,3],0) -3*Porb(2,[1,1],2)/(1-t^2); // 0 each of the 3 1/4(2,3,3) points eats -1/2.1/4, making 3 > d:=28;L:=[1,1,3,9,14]; (1-t^d)/denom(L)-PI(L)-Qorb(9,[1,3,5],0); // 3*t^3/(t^6 - 3*t^5 + 3*t^4 - 2*t^3 + 3*t^2 - 3*t + 1), leaving no (1-t^3)^2 in denom > d:=33;L:=[1,1,4,11,16]; (1-t^d)/denom(L)-PI(L)-Qorb(16,[1,4,11],0)-0*Porb(2,[1,1],2)/(1-t^2); // 2*t^3/(t^6 - 4*t^5 + 7*t^4 - 8*t^3 + 7*t^2 - 4*t + 1), leaving no (1-t^4)^2 in denom The disaster: d:=25;L:=[1,1,5,6,12]; (1-t^d)/denom(L)-PI(L)-Qorb(12,[1,5,6],0)-Porb(6,[1,5],6)/(1-t^6) -4/3*(t^4+t^5)/denom([1,1,1,6])-5/3*t^3/denom([1,1,1,3])+1/3*Porb(2,[1,1],2)/(1-t^2); // 0 > -4/3*(t^4+t^5)/denom([1,1,1,6]) -5/3*t^3/denom([1,1,1,3]) +1/3*Porb(2,[1,1],2)/(1-t^2); (-2*t^7 - 3*t^6 - 3*t^5 - 3*t^4 - 2*t^3)/(t^10 - 2*t^9 + 2*t^7 - t^6 - t^4 + 2*t^3 - 2*t + 1) // that is, denom([1,1,2,6]) d:=30;L:=[1,5,6,6,12]; (1-t^d)/denom(L)-PI(L)-Qorb(12,[1,5,6],0)-3*Porb(6,[1,5],6)/(1-t^6) -4/3*(t^4+t^5)/denom([1,1,1,6])-5/3*t^3/denom([1,1,1,3])+1/3*Porb(2,[1,1],2)/(1-t^2); // 0 ----------------- Now I do surfaces ----------------- S16 in PP(1,4,5,5) has k=1 so c = 4, PI := (t^4-2*t^3+t^2-2*t+1)/(1-t)^3; > d:=16;L:=[1,4,5,5];(1-t^d)/denom(L)-(1-2*t+t^2-2*t^3+t^4)/(1-t)^3 - Porb(5,[4],6)/(1-t^5); (-2*t^5 - 2*t^4 - 2*t^3)/(t^7 - 2*t^6 + t^5 - t^2 + 2*t - 1) One problem is that since there is only one 1, the tangent terms along the 1/5(4) curve boil down to a_3(z_1,z_2)x, so there are 3 additional nonquasismooth singularities to worry about. The above answer 2*(t^3+t^4+t^5)/denom(1,1,5) possibly already takes this into account. S20 in PP(4,5,5,5) k=1, c=4, PI = (t^4-3*t^3+3*t^2-3*t+1)/(1-t)^3; > d:=20;L:=[4,5,5,5];(1-t^d)/denom(L)-(t^4-3*t^3+3*t^2-3*t+1)/(1-t)^3-4*Porb(5,[4],6)/(1-t^5); (-2*t^5 - 2*t^4 - 2*t^3)/(t^7 - 2*t^6 + t^5 - t^2 + 2*t - 1) S50 in PP(5,9,10,25) > d:=50;L:=[5,9,10,25];(1-t^d)/denom(L)-(t^4-3*t^3+3*t^2-3*t+1)/(1-t)^3-Porb(9,[1,7],1)-Porb(5,[4],6)/(1-t^5); (-t^5 - t^4 - t^3)/(t^7 - 2*t^6 + t^5 - t^2 + 2*t - 1) Now get 1*standard factor (t^3+t^4+t^5)/denom(1,1,5) //======================================================== // This is the point contribution to orbifold RR, including embedded points function Qorb(r,LL,k) if (k + &+[i : i in LL]) mod r ne 0 then error "Error: Canonical weight not compatible"; end if; n := #LL; h := Degree(GCD(1-t^r, denom(LL))); // degree of GCD(A,B) -- simpler calc? l := Floor((k+n+1)/2+h); // If l < 0 we need a kludge to avoid programming Laurent polynomials properly de := Maximum(0,Ceiling(-l/r)); m := l + de*r; A:=(t^r-1) div (t-1); B:= &*[(1-t^i) div (1-t) : i in LL]; H,al_throwaway,be:=XGCD(A,t^m*B); return t^m*be/(H*(1-t)^n*(1-t^r)*t^(de*r)); end function; // the old stuff before I had Qorb in place S81 in PP(5,19,25,31) experimenting suggests the right value of Qorb(25,[5,19],1) comes by putting l = 6 or 7 in function, giving (-t^25 - t^19 - t^13 - t^7)/(t^31 - t^30 - t^26 + t^25 - t^6 + t^5 + t - 1), that is, (t^7+t^13+t^19+t^25)/denom([1,5,25]) > d:=81;L:=[5,19,25,31];(1-t^d)/denom(L)-(t^4-3*t^3+3*t^2-3*t+1)/(1-t)^3-Porb(\ 19,[6,12],1)-Porb(31,[5,25],1)-0*Porb(5,[4],6)/(1-t^5)-(t^7+t^13+t^19+t^25)/de\ nom([1,5,25]); (-2*t^5 - 2*t^4 - 2*t^3)/(t^7 - 2*t^6 + t^5 - t^2 + 2*t - 1) 95 in PP(5,19,25,45) has 1/25(5,19) and 1/45(19,25) sings, so is a test for Qorb function. d:=95;L:=[5,19,25,45];(1-t^d)/denom(L)-(t^4-3*t^3+3*t^2-3*t+1)/(1-t)^3; gives a jolly complicated function same - Qorb(25,[5,19],1) - Qorb(45,[19,25],1) gives very neat short formula (but wrong) so can use this for testing function Qorb. any of l := 3,4,5,6,7,8,9,19,11 in the function gives (-t^41 + t^39 - t^36 + t^32 - t^31 - t^21 + t^20 - t^16 + t^13 - t^11) /(t^51 - t^50 - t^46 + t^45 - t^6 + t^5 + t - 1) // now the right versions: > d:=81;L:=[5,19,25,31];(1-t^d)/denom(L)-(t^4-3*t^3+3*t^2-3*t+1)/(1-t)^3-Porb(\ 19,[6,12],1)-Porb(31,[5,25],1)-Qorb(25,[5,19],1); (-2*t^5 - 2*t^4 - 2*t^3)/(t^7 - 2*t^6 + t^5 - t^2 + 2*t - 1) > d:=100;L:=[5,19,25,50];(1-t^d)/denom(L)-(t^4-3*t^3+3*t^2-3*t+1)/(1-t)^3-Porb\ (19,[6,12],1)-2*Qorb(25,[5,19],1); (-2*t^5 - 2*t^4 - 2*t^3)/(t^7 - 2*t^6 + t^5 - t^2 + 2*t - 1) > d:=95;L:=[5,19,25,45];(1-t^d)/denom(L)-(t^4-3*t^3+3*t^2-3*t+1)/(1-t)^3-Qorb(\ 25,[5,19],1)-Qorb(45,[19,25],1)-Porb(5,[4],6)/(1-t^5); (-2*t^5 - t^4 - 2*t^3)/(t^7 - 2*t^6 + t^5 - t^2 + 2*t - 1) > d:=140;L:=[5,19,45,70];(1-t^d)/denom(L)-(t^4-3*t^3+3*t^2-3*t+1)/(1-t)^3 -Porb(19,[5,13],1)-Qorb(45,[19,25],1)-Porb(5,[4],6)/(1-t^5); (-t^5 - t^3)/(t^7 - 2*t^6 + t^5 - t^2 + 2*t - 1) These last two cases are minor disasters since they are not of the standard form. They have some kind of N-terms. -------------------------------- Now I can do arbitrary PP(a,b,c) with no common factor in a,b,c -------------------------------- PP(1,5,15) > L:=[1,5,15]; 1/denom(L)-Qorb(15,[1,5],-21); 0 > L:=[1,15,25]; 1/denom(L)-Qorb(15,[1,10],-41)-Qorb(25,[1,15],-41); 0 > for i in [1..100] do for> L:=[1,15,15+5*i]; 1/denom(L)-Qorb(15,[1,5*(i mod 3)],-31-5*i)-Qorb(15+5*i,[1,15],-31-5*i); for> end for; // 0 all the way > for i in [1..4] do > L:=[2,15,15+10*i]; 1/denom(L)-Qorb(15,[2,10*(i mod 3)],-32-10*i) -Qorb(15+10*i,[2,15],-32-10*i)-Porb(2,[1,1],-32-10*i); > end for; i:=1;L:=[2,15,15+10*i]; 1/denom(L)-Qorb(15,[2,10*(i mod 3)],-32-10*i) -Qorb(15+10*i,[2,15],-32-10*i)-Porb(2,[1,1],-32-10*i); need to add 1*Porb(5,[2],-37)/(1-t^5) to get i:=1;L:=[2,15,15+10*i]; 1/denom(L)-Qorb(15,[2,10*(i mod 3)],-32-10*i) -Qorb(15+10*i,[2,15],-32-10*i)-Porb(2,[1,1],-32-10*i)+Porb(5,[2],-27-10*i)/(1-t^5); (-t^3 - 1)/(t^26 - 2*t^25 + t^24 - t^21 + 2*t^20 - t^19) should be read as (t^-19 + t^-16)/denom(1,1,5) > PartialFractionDecomposition($2*t^19); [ , , , ] Same thing for i = 2 needs - Porb(5,[2],-47)/(1-t^5, giving i:=2;L:=[2,15,15+10*i]; 1/denom(L)-Qorb(15,[2,10*(i mod 3)],-32-10*i) -Qorb(15+10*i,[2,15],-32-10*i)-Porb(2,[1,1],-32-10*i)-Porb(5,[2],-27-10*i)/(1-t^5); (t^3 + 1)/(t^31 - 2*t^30 + t^29 - t^26 + 2*t^25 - t^24) should be read as (-t^-24 - t^-21)/denom(1,1,5) > PartialFractionDecomposition($2*t^24); [ , , , ] When i = 3 the HCF is 15 and you need +1 i:=3;L:=[2,15,15+10*i]; 1/denom(L)-Qorb(15,[2,10*(i mod 3)],-32-10*i) -Qorb(15+10*i,[2,15],-32-10*i)-Porb(2,[1,1],-32-10*i)+1*Porb(15,[2],-17-10*i)/(1-t^15); this is t^-29 times (t^13 + t^11 + t^9 + t^7 + t^6 + t^4 + t^2 + 1)/denom(1,1,15) i:=4;L:=[2,15,15+10*i]; 1/denom(L)-Qorb(15,[2,10*(i mod 3)],-32-10*i) -Qorb(15+10*i,[2,15],-32-10*i)-Porb(2,[1,1],-32-10*i)-Porb(5,[2],-27-10*i)/(1-t^5); gives (-1-t^3)/(denom([1,1,5])*t^34); i:=5;L:=[2,15,15+10*i]; 1/denom(L)-Qorb(15,[2,10*(i mod 3)],-32-10*i) -Qorb(15+10*i,[2,15],-32-10*i)-Porb(2,[1,1],-32-10*i)-(-1)^i*Porb(5,[2],-27-10*i)/(1-t^5) eq -(-1)^i*(1+t^3)/(denom([1,1,5])*t^(14+5*i)); for i in [1..100] do if GCD(3,i) eq 1 then L:=[2,15,15+10*i]; 1/denom(L)-Qorb(15,[2,10*(i mod 3)],-32-10*i)-Qorb(15+10*i,[2,15],-32-10*i) -Porb(2,[1,1],-32-10*i)-(-1)^i*Porb(5,[2],-27-10*i)/(1-t^5) eq -(-1)^i*(1+t^3)/(denom([1,1,5])*t^(14+5*i)); end if; end for; // true all the way i:=1;L:=[2,15,15*i];1/denom(L)-Qorb(15*i,[2,15],-17-15*i); 1/denom(2,15,15), of course for j in [1..10] do i:=2*j-1;L:=[2,15,15*i];1/denom(L)-Qorb(15*i,[2,15],-17-15*i) -Porb(2,[1,1],-17-15*i)+(-1)^j*Porb(15,[2],-2-15*i)/(1-t^15) eq (-1)^j*(t^13 + t^11 + t^9 + t^7 + t^6 + t^4 + t^2 + 1)/(t^(15*j-1)*denom([1,1,15])); end for; // true all the way PP(a,b,c) suppose a,b div by 3 and a,c div by 7, PP(21,3*a,7*b) P(21,3,7) a:=1; b:=1;L:=[21,3*a,7*b];k:=-&+[i: i in L]; 1/denom(L)-Qorb(21,[3*a,7*b],k)-Qorb(7*b,[21,3*a],k); // 0 P(21,3,14) a:=1; b:=2;L:=[21,3*a,7*b];k:=-&+[i: i in L]; 1/denom(L)-Qorb(21,[3*a,7*b],k)-Qorb(7*b,[21,3*a],k) +0*Porb(7,[3*a],k+7)/(1-t^7)-0*Porb(3,[7*b],k+3)/(1-t^3); a:=2; b:=1;L:=[21,3*a,7*b];k:=-&+[i: i in L]; 1/denom(L)-Qorb(21,[3*a,7*b],k)-Qorb(7*b,[21,3*a],k) +0*Porb(7,[3*a],k+7)/(1-t^7)-0*Porb(3,[7*b],k+3)/(1-t^3); a:=3; b:=1;L:=[21,3*a,7*b];k:=-&+[i: i in L]; 1/denom(L)-Qorb(21,[3*a,7*b],k)-Qorb(7*b,[21,3*a],k) -Porb(7,[3*a],k+7)/(1-t^7)-0*Porb(3,[7*b],k+3)/(1-t^3); a:=1; b:=1; L:=[6,2*a,3*b]; k:=-&+[i : i in L]; 1/denom(L)-Qorb(6,[2*a,3*b],k)-Qorb(2,[6,3*b],k)-Qorb(3,[6,2*a],k) -0*Porb(2,[3*b],k+2)/(1-t^2)-0*Porb(3,[2*a],k+3)/(1-t^3); S:= (-2/21*t^5 - 2/21*t^4 - 4/21*t^3 - 5/21*t^2 - 2/21*t + 1/21)/(1-t^7); (5/21*t^5 + 8/21*t^4 + 3/7*t^3 + 8/21*t^2 + 5/21*t)*(1-t); (1/7*t^5 + 2/7*t^4 + 5/21*t^3 + 1/7*t^2 + 1/7*t + 1/21)*(1-t)/(1-t^7); function AB(r,L) K := CyclotomicField(r); return 1/r * 1/(1-t^r) * &+[ Q! &+[(ep^(-i*j)-1)/&*[1-ep^(-a*j) : a in L] : j in [1..r-1] | &and[(a*j) mod r ne 0 : a in L]] * t^i : i in [1..r-1]]; end function; Notes: Don't really need k (it just permutes the mu_r roots in L) Interesting case are AB(15,[1,9],1), AB(15,[1,3],1) etc 1/15(3,2) is an embedded point on a curve of 1/3(2) > AB(15,[3,2],14); (2/5*t^12 + 2/5*t^9 - 2/5*t^8 + 2/5*t^7 - 1/5*t^5 + 1/5*t^4 + 1/5*t^3 - 2/5*t^2 + 2/5*t)/(t^13 - t^12 + t^10 - t^9 + t^7 - t^6 + t^4 - t^3 + t - 1) > PartialFractionDecomposition($1); [ , , ] > Qorb(15,[3,2],10); (t^20 - t^19 + t^18 + t^11 - t^10 + t^9)/(t^19 - t^18 - t^16 + t^15 - t^4 + t^3 + t - 1) > PartialFractionDecomposition($1); [ <1, 1, t>, , , , , , , ] > AB(15,[2,3],14) eq 1/denom([2,3,15]) - Porb(2,[3,15],-20) - 1/5*Porb(3,[2],-17)/(1-t^3) + (1/45*t-1/45) * 1/((1+t+t^2)*t^8)+8/15*1/((1-t)^3*t^8) + 4/15/((1-t)^2*t^8) + 1/45*1/(t^8*(1-t)) + (2/5*t^7 +2/5*t^6 + 2/5*t^5 + 2/5*t^4 + 2/5*t^2 + 1/5)/t^8; true > AB(15,[2,3],14) + 1/5*Porb(3,[2],-17)/(1-t^3)