Propiedades físicas de un fluido de perforación

In this section, we will introduce the K3 surfaces which have a BHK Mirror, so called surfaces of BHK type. We will start by introducing weighted Delsarte surfaces that are K3 surfaces and then move to quotienting by certain symplectic quotients.

4.2.1

Weighted Delsarte Surfaces

Let k be an algebraically closed field. Take FA to be a sum of four monomials in four variables FA:= 3 X i=0 3 Y j=0 xaij j .

We require that the polynomial FA satisfy three conditions:

2. there exists positive integers qi and m so that, for all i, m equals the sum

P3

j=0aijqj; and

3. when the polynomial is viewed as a mapFA :kn+1 →k, its only critical point is the origin.

These conditions are invertibility, quasihomogeneity and quasismoothness, re- spectively. The zero locus of the polynomial FA cuts out a well-defined degree m hypersurface XA :={FA= 0} in the weighted-projective space WP3(q0, . . . , q3).

Remark 4.2.1. When qi = 1 for all i, the hypersurface XA⊆P3 is called a Delsarte surface. For this reason, the hypersurface XAis called a weighted Delsarte surface. When in characteristic zero, condition (3) above implies that the singular locus of the hypersurfaceXAis exactly the singular locus of the weighted projective space intersected with the hypersurface itself, i.e.,

Sing(XA) =XA∩Sing(WP3(q0, q1, q2, q3).

Remark 4.2.2 ([21]). Recall that there is an explicit description of the singular locus of a weighted projective space. A point x = (x0, x1, x2, x3) ∈ WP3(q0, q1, q2, q3) is in the singular locus of the weighted projective space if and only if the quantity gcd(qi :i∈I(x)) is greater than one, where the set I(x) is {i:xi 6= 0}.

When over an arbitrary algebraically closed field, we will add an additional condition to our hypersurface:

Definition 4.2.3 ([20]). We sayXA is in general position if

codimXA(XA∩Sing(WP

3

(q0, q1, q2, q3))) = 2. (4.2.1)

Lemma 4.2.4 ([20]). Let XA be a quasismooth hypersurface in general position in

W_P3_(q

0, q1, q2, q3), then

Sing(XA) = XA∩Sing(WP3(q0, q1, q2, q3)).

From here on, we will assume that XA is in general position, if over a field of positive characteristic. Given a weighted Delsarte surface XA, we can now calculate the canonical class of its (minimal) resolution ˜XA99KXA to be

ω_X˜_A ∼=O_X˜_A(m−q0−q1−q2−q3).

So, XA is a (possibly singular) K3 surface if P3_i=0qi =m, or, equivalently the sum

P3

i=0a

ij _{equals 1.}

4.2.2

Symplectic Group Actions

In this section, we introduce the symplectic group actions on a weighted Delsarte surface XA that are outlined in the Berglund-H¨ubsch-Krawitz mirror construction. We first will start by defining what we mean by symplectic group actions over fields that are not the complex numbers.

Definition 4.2.5 ([27]). Let X be a normal surface over k. Let G be a finite group of k-automorphisms of X. Denote by Y the quotient surface X/G and by

1. A surface X is said to be an orbifold K3 surface if the canonical sheaf ωX is isomorphic to the structure sheafOX, the first cohomology classH1(X,OX) of the structure sheaf vanishes, and the canonical sheaf of the minimal resolution

σ : ˜X → X is just the pullback of the canonical sheaf of X along σ, i.e.,

ω_X˜ ∼=σ∗(ωX).

2. We say that the quotient map π :X →Y contains no wild codimension one ramification if the characteristic of k does not divide the order of the inertia group of the map π at every prime divisor ofX.

3. The group action of G on X is called symplectic if every element of G fixes the nowhere vanishing 2-form on X, i.e., g∗ωX =ωX for all g ∈G.

Lemma 4.2.6. Assume π : X → Y as above to contain no wild codimension one ramification. Then the canonical sheaf of Y ωY is isomorphic to (π∗ωX)G. If

additionally, the surface X is a K3, then ωY ∼=OY.

We now would like to give a few facts about the above objects:

Remark 4.2.7. If the characteristic of the field k does not divide the order of the group G, then the map π contains no wild codimension one ramification.

Remark 4.2.8. When working over a field of positive characteristic, there exists examples of a K3 surface X and finite group G such that G is a symplectic group acting onX and the quotientX/Gis not an orbifold K3 surface. (See Example 2.8 of [27]).

Consider the group of automorphisms of the polynomial FA, denoted Aut(FA): Aut(FA) := (λ0, λ1, λ2, λ3)∈(k∗)4 FA(λ0x0, λ1x1, λ2x2, λ3x3) = FA(x0, x1, x2, x3)

We can describe the elements of Aut(FA) easily as being generated by the elements of the torus (_C∗)4 _{generated by}

ρj = (e2πia

0j

, e2πia1j, e2πia2j, e2πia3j).

This group does not act symplectically on the hypersurfaceXAbut it has a subgroup

SL(FA) that does, which can be described as:

SL(FA) := ( (λ0, λ1, λ2, λ3)∈Aut(FA) 3 Y i=0 λi = 1 ) .

The group SL(FA) contains a subgroup, the exponential grading operator group

JFA, that acts trivially on the hypersurfaceXA. We can describe this group as:

JFA :={(λ

q0_{, λ}q1_{, λ}q2_{, λ}q3_)∈SL(F

A)|λ ∈k∗}.

Take a group G so that JFA ⊆ G ⊆ SL(FA). Then the quotient ¯G := G/JFA is

a subgroup of the (nontrivial) automorphisms of XA that leave the nonvanishing 2-form invariant. We then take the orbifold

ZA,G=XA/G.¯

Any orbifold K3 surface that can be written as ZA,G for an appropriate choice of

A and G are defined to be K3 surfaces of BHK type. In the next section, we will describe what these orbifolds look like birationally and then compute their Picard numbers, which will relate to their BHK mirrors.

4.2.3

The Berglund-H¨ubsch-Krawitz Mirror

In this section, we construct the BHK mirror to the Calabi-Yau orbifoldZA,Gdefined above. Take the polynomial

F_AT = 3 X i=0 3 Y j=0 Xaji j . (4.2.2)

It is quasihomogeneous because there exist positive integers ri :=

P

jbji so that

F_AT(λr0X₀, . . . , λr3X₃) = λmF_AT(X₀, . . . , X₃), (4.2.3)

for all λ ∈ k∗. Note that the polynomial FAT cuts out a well-defined Calabi-Yau

hypersurface X_AT ⊆ W_Pn(r₀, . . . , r₃). Define the diagonal automorphism group,

Aut(F_AT), analogously to Aut(F_A). The group Aut(F_AT) is generated by ρT_i :=

diag(e2πibij/d₎3

j=0∈(k∗)4. Define the dual group GT relative to Gto be

GT := ( _n Y i=0 (ρT_i )si si ∈Z, where n Y i=0 xsi i is G-invariant ) ⊆Aut(F_AT). (4.2.4)

Note that the dual group GT sits between JF_AT and SL(FAT) (for details, see

[30]). Define the group ˜GT _{:= G}T_/J

F_AT. We have a well-defined K3 orbifold

Z_AT_,GT := X_AT/G˜T ⊂ W_Pn(r₀, . . . , r₃)/G˜T. The K3 orbifold Z_AT_,GT is the BHK

mirror to ZA,G.