4.4. Efecto de la harina de variedades de quinua en las características del pan sin gluten
4.4.1. Volumen y altura
In this section, we give some examples of our parsing theorem and show how to use it to construct orbifolds. Recall that Xd represents a general hypersurface defined
by a homogeneous polynomial of degree d and Xd1,d2 the complete intersection of two general hypersurfaces of degreed1, d2.
Example 4.7.1. Consider the following three orbifolds who have trivial canonical sheaves. We choose the polarization to be O(1) for all of them.
• S5 = Stack[x, y, z, t]/(f5)⊂P(1,1,1,2)with an orbifold point Q= (0,0,0,1)
of type 1 2(1,1);
• S7 = Stack[x, y, z, t]/(f7) ⊂ P(1,1,2,3) with orbifold points Q1 = (0,0,1,0)
of type 12(1,1)and Q2 = (0,0,0,1)of type 13(1,2);
• S11= Stack[x, y, z, t]/(f11)⊂P(1,2,3,5)with orbifold points Q1 = (0,1,0,0)
of type 12(1,1), Q2 = (0,0,1,0) of type 13(1,2) and Q3 = (0,0,0,1) of type 1
5(2,3);
where fi is a general polynomial of degree i in the corresponding graded ring. All
three surfaces have kSi = 0 and c = 3. The Hilbert series can be parsed into
PSi(t) =PI+ P
Q∈BiPorb,Q(t), in which PI(t) can be calculated as in Lemma 4.6.1
andPorb can be calculated using Program 4.5.7. This gives PS5(t) = PI(t) +Porb,Q(t) = 1 +t3 (1−t)3 + t2 (1−t)2(1−t2); PS7(t) = PI(t) +Porb,Q1(t) +Porb,Q2(t) = 1−t−t2+t3 (1−t)3 + t2 (1−t)2(1−t2) + t2+t3 (1−t)2(1−t3); PS11(t) = PI(t) +Porb,Q1(t) +Porb,Q2(t) +Porb,Q3(t)
= 1−2t−2t2+t3 (1−t)3 + t2 (1−t)2(1−t2) + t2+t3 (1−t)2(1−t3)+ 2t2+t3+t4+ 2t5 (1−t)2(1−t5) . One notices the same contribution from orbifold points of the same type when the orbifolds have the same canonical weight. Next we want to use the above data to find more orbifolds with certain orbifold points. Since the orbifold term only depends on the orbifold type of the point and the canonical weight, in the rest of
this section we will use notationPorb(1r(a1, . . . , an)) to denote the orbifold term for
the point of type 1
r(a1, . . . , an) when the canonical weight is clear.
Example 4.7.2. Based on the first example withS5 ⊂P(1,1,1,2), if we want one more orbifold point of the same type and keep the other invariants unchanged, then we have the following data:
• P0 = 1, P1= 3;
• 2 points of type 12(1,1).
We havePI = (1 +t3)/(1−t)3 and Porb =t2/(1−t)2(1−t2) as in the last example.
Therefore we have
P(t) =PI+ 2Porb(12(1,1)) = 1−t
3−t4+t7 (1−t)3(1−t2)2,
which suggests that such an orbifold can be a complete intersectionXf,g withf, gtwo general homogeneous polynomial of degree 3 and4 respectively inside P(1,1,1,2,2). We can easily analyze that it has two orbifold points at the intersection of {g = 0}
withP(2,2).
Next, based onS11⊂P(1,2,3,5), we want to construct an orbifold with
• P0 = 1, P1= 1;
• 2 points of type 13(1,2);
• 1 point of type 15(2,3). Then similarly one can calculate
PI = 1−2t−2t2+t3 (1−t)3 , and then P(t) =PI(t) + 2Porb(13(1,2)) +Porb(15(2,3)) = 1−t 6−t8+t14 (1−t)(1−t2)(1−t3)2(1−t5).
Thus, this orbifold with the required property can be given by a complete intersection
with no other orbifold points, then one has
P(t) =PI(t) + 2Porb(15(2,3)) = 1−t
6−t10+t16
(1−t)(1−t2)(1−t3)(1−t5)2,
which gives an orbifold with trivial canonical sheaf and two orbifold points as re- quired.
The above example just gives us some idea about how the combination of certain PI and Porb may give some reasonable Hilbert series P(t) which helps us to find the required orbifolds. It is also very efficient to construct some higher dimensional canonical orbifolds. In the following, we give some examples, where we construct canonical 3-folds (canonical weight equals 1).
To do experiments, we give a program for the initial term. Suppose we want an orbifold of dimension n with canonical weight kX with P0, P1, . . . , Pbc
2c as the
firstbc
2cplurigenera. The following program gives us the initial term.
Program 4.7.2. function initial(L,k,n) f:=&+[L[i]*t^(i-1): i in [1..#L]]; pp:=R!(f*(1-t)^(n+1));
c:=k+n+1;
if IsEven(c) eq true then
return (&+[Coefficient(pp, i )*(t^i+t^(c-i)):i in [0..c div 2-1]]+ Coefficient(pp,c div 2)*t^(Floor(c/2)))/(1-t)^(n+1);
else
return &+[Coefficient(pp,i)*(t^i+t^(c-i)):i in [0..Floor(c/2)]] /(1-t)^(n+1);
end if;
end function;
Example 4.7.3. Suppose we want to construct a canonical3fold withP0= 1, P1 = 2, P2 = 4. Suppose we can have 0, 1 or 2 of the orbifold points of types 1
2(1,1,1), 1
3(1,2,2) and 15(1,1,2). Using Program 4.5.7, we can calculate Porb(t) for each of
these points. Then we can search using the following MAGMA program:
q1:= Qorb (5,[1,1,2],1); q2:= Qorb (3,[1,2,2],1); q3:= Qorb (2,[1,1,1],1);
for i,j, k in [0..2] do
p:= initial ([1,2,4],1,3)+i*q1+j*q2+k*q3; p* &*[(1-t^i): i in [1,1,2,3,5]]; [i,j,k]; end for;
Analyzing the output of the above program, we have the following canonical
3-folds:
• X13⊂P(1,1,2,3,5)has one orbifold point for each type above.
• X16⊂P(1,1,2,3,8)has 1 point of orbifold type 13(1,2,2) and 2 points of type 1
2(1,1,1).
• X20 ⊂ P(1,1,2,5,10) has 2 points of orbifold type 15(1,2,2) and 2 points of
type 12(1,1,1).
• X6,10⊂P(1,1,2,3,3,5)has 2 points of orbifold type 13(1,2,2).
• X8,9 ⊂ P(1,1,2,3,4,5) has 1 point of type 15(1,1,2) and 2 points of type 1
2(1,1,1).
• X8,10 ⊂ P(1,1,2,3,5,5) has 1 point of type 2 points of type 1
5(1,2,2) and 1
3(1,2,2).
These are all the complete intersections and hypersurfaces under this condition. There are also codimension3 outputs. For example, one of the outputs tells us that a canonical3-fold with2points of type 15(1,1,2)and one point of type 12(1,1,1)gives rise to the Hilbert series
P(t) = 1−2t8−2t9−t10+t12+ 2t13+ 2t14−t22 (1−t)2(1−t2)(1−t3)(1−t4)(1−t5)2 ,
which suggests that the orbifold we want can be given as a codimension3 substack inP(1,1,2,3,4,5,5)defined by the4×4 Pfaffians of a5×5 skew symmetric matrix with the following degree in each entry:
6 5 5 4 5 5 4 4 3 3 .
Let x1, x2, y, z, t, s1, s2 be the coordinates of P(1,1,2,3,4,5,5). Then we can fill out
each entry of the matrix with a general polynomial of the indicated degrees to make sure that we get the right orbifold points. For example, we fill in as follows:
z2 s 1 −s2 t x51+s2 −x52−s1 y2+t y2−t z+x3 2 z+x3 1 ,
which gives 5 equations, that is, the 4×4 Pfaffians
pf1 = x81+x82+zs1+zs2+y4−t2+· · · pf2 = zs1+zs2+ty2−t2+s1x31+s2x32 pf3 = z3−s1t+s2t−x52t+z+x31−tx52 pf4 = z3+s1t−s2t+x51t+s1y2+· · · pf5 = s21−s22−z2t+· · ·
We can check that this gives a quasismooth orbifold with 2 orbifold points of type
1
5(1,1,2)alongP(5,5)and1 point of type 12(1,1,1) alongP(2,4). We can also vary
the coefficients of the polynomials in the matrix, and in this way, we find a family of codimension 3 canonical 3-folds with required properties.
Similarly there are other outputs giving us candidates for codimension 3
canonical 3-folds. We donot list them one by one, but just remark that with help of computer programs, this method of finding varieties with certain invariants is quite efficient (compare with [ABR02] and [Rei00]). For the codimension 4 guys we need to use the unprojection method developed in [PR04a], [PR04b].