FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 22 definitions, 7 theorems, 7 propositions, 4 lemmas, 3 corollaries, 1 example, 3 constructions, 3 remarks and 1 convention.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
In this chapter \(X\) is a K3 surface and we write \(\Lambda\) instead of \(\Lambda _ { \mathsf { K 3 } }\) for the K3 lattice.
Proposition 19.1.1
Proposition 19.1.1. All K3 surfaces \(X\) are K¨ahler and are mutually diffeomorphic. The Betti numbers are \(b _ { 1 } ( X ) = b _ { 3 } ( X ) = 0\) , \(b _ { 2 } ( X ) ~ = ~ 2 2\) , and the Hodge numbers are \(h ^ { 1 , 0 } ( X ) = h ^ { 0 , 1 } ( X ) = 0\) , \(h ^ { 2 , 0 } ( X ) = 1\) and \(h ^ { 1 , 1 } ( X ) = 2 0\) . The intersection lattice \(H ^ { 2 } ( X , \mathbb { Z } )\) is isometric to the \(K \mathcal { 3 }\) -lattice \[ \Lambda : = U ^ { \oplus 3 } \oplus E _ { 8 } ( - 1 ) ^ { \oplus 2 } . \]
Proposition 19.1.2
Proposition 19.1.2. Let \(D\) be a divisor on \(X\) . 1. If \(D \cdot D \geq - 2\) either \(D\) or \(- D\) is effective. 2. If ?? is irreducible, then \(D \cdot D \geq - 2\) and if equality holds \(D\) is a nodal curve, that is, a smooth rational curve. 3. ?? is ample if and only if \(D \cdot D > 0\) and \(D \cdot D ^ { \prime } > 0\) for all nodal curves \(D ^ { \prime }\) .
Definition / construction / notation
For brevity, the classes of effective, respectively ample divisors in the N´eron– Severi lattice \(\mathsf { N S } ( X )\) are called effective, respectively ample classes. Likewise, the class of a nodal curve is called a nodal class. Observe that nodal classes are roots in \(\mathsf { N S } ( X )\) , but not conversely. In fact, \(\left( - 2 \right)\) -roots in the N´eron–Severi lattice are classes of divisors \(D\) with \(D \cdot D = - 2\) which are not necessarily effective or irreducible. Reflections in \(\left( - 2 \right)\) -roots in \(\mathsf { N S } ( X )\) generate its Weyl group. Since the corresponding hyperplane reflections extend to \(H ^ { 2 } ( X , \mathbb { Z } )\) , this group can be considered as a subgroup of \(\mathsf { O } ( H ^ { 2 } ( X , \mathbb { Z } ) )\) . We call it the Weyl group of \(X\) : \[ \mathsf { W } ^ { - } ( X ) : = \mathsf { W } ^ { - } ( \mathsf { N S } ( X ) ) \subset \mathsf { O } ( H ^ { 2 } ( X , \mathbb { Z } ) ) . \]
Construction / comparison
The Weyl group of a K3 surface is a normal subgroup of the orthogonal group of its integral second cohomology.
Definition / construction
The roots in the Néron–Severi lattice of a K3 surface have a canonical partition \(\Delta(NS(X))=\Delta_X^+\sqcup(-\Delta_X^+)\), where \(\Delta_X^+\) is the set of effective roots.
Definition / construction / notation
The Hodge index theorem B.2.3 states that the intersection product on \(H _ { \mathbb { R } } ^ { 1 , 1 } ( X ) =\) \(H ^ { 1 , 1 } ( X ) \cap H ^ { 2 } ( X , \mathbb { R } )\) has signature \(( 1 , h ^ { 1 , 1 } ( X ) - 1 )\) , and so it gives \(H _ { \mathbb { R } } ^ { 1 , 1 } ( X )\) the structure of a hyperbolic vector space containing \(\mathsf { N S } ( X )\) . As explained in Appendix B.2, there is a preferred component \(\complement _ { X }\) of the “light cone” \(\{ x \in H _ { \mathbb { R } } ^ { 1 , 1 } ( X ) \mid x \cdot x > 0 \}\) , namely the one that contains the K¨ahler classes and which is called the positive cone. The group \(\mathsf { W } ^ { - } ( X )\) acts on \(H _ { \mathbb { R } } ^ { 1 , 1 } ( X )\) and by (16.3) it preserves the positive cone.
Definition / construction
The Kähler cone of a K3 surface is the subcone of the positive cone consisting of classes pairing positively with every nodal class.
Definition / construction
For a projective K3 surface, the cone spanned by ample classes is the intersection of the Kähler cone with \(NS(X)\otimes\mathbf R\).
Definition / construction / notation
This cone is called the ample cone of \(X\) . The ample classes are precisely the integral classes of the ample cone.
Proposition 19.1.3
Proposition 19.1.3. 1. The closure of the K¨ahler cone is a fundamental domain for the action of the Weyl group \(\mathsf { W } ^ { - } ( X )\) (cf. (19.1)) on \(C _ { X }\) . 2. If \(X\) is projective, the closure of the ample cone is a fundamental domain for the action of the Weyl group \({ \mathsf { W } } ^ { - } ( X )\) on the intersection of the positive cone with \({ \mathsf { N S } } ( X ) \otimes \mathbb { R }\) . 3. The set of nodal classes determines the K¨ahler cone and if \(X\) is projective, these nodal classes determine the ample cone. 4. The choice of a K¨ahler class determines the entire K¨ahler cone and in the projective case, the choice of an ample class determines the entire ample cone.
Theorem 19.2.1
Theorem 19.2.1. 1. Two \(K \mathcal { 3 }\) -surfaces are isomorphic if and only if there are markings such that the corresponding period points are the same (”injectivity of the period map” – or – ”Torelli theorem”). 2. All points of the period domain are period points (”surjectivity of the period map”).
Theorem 19.2.2
Theorem 19.2.2. Let \(X\) be a \(K \mathcal { 3 }\) surface and ?? an isometry of \(H ^ { 2 } ( X , \mathbb { Z } )\) . Then there exists a unique automorphism of \(X\) inducing ?? if and only if the following conditions hold simultaneously: • the complexification ??ℂ of ?? preserves the Hodge decomposition, or, equiva lently, ??ℂ preserves the line spanned by the holomorphic 2-form. • ??ℝ preserves the K¨ahler cone, or equivalently (by Proposition 19.1.3), sends some K¨ahler class to a K¨ahler class.
Corollary 19.2.3
Corollary 19.2.3. An isometry of \(H ^ { 2 } ( X , \mathbb { Z } )\) preserving the Hodge decomposition is, up to sign and (left or right) multiplication with an element of \(\mathsf { W } ^ { - } ( X )\) , induced by a unique automorphism of \(X\) .
Lemma 19.2.4
Lemma 19.2.4. For a \(K \mathcal { 3 }\) surface \(X\) there is an injection \[ \operatorname { A u t } ( X ) \hookrightarrow \operatorname { O } ( H ^ { 2 } ( X , \mathbb { Z } ) ) , \quad g \mapsto g ^ { * } . \]
Construction / comparison
Under the faithful action on cohomology, \(Aut(X)\) identifies with the Hodge isometries preserving the Kähler cone; equivalently, Hodge isometries are obtained modulo sign and the Weyl-group action as in the displayed formula.
Lemma 19.2.5
Lemma 19.2.5. The marked \(K \mathcal { 3 }\) -surfaces \(( X , \varphi )\) , \(( X , \pm \varphi \circ w )\) , \(w \in \mathsf { W } ^ { - } ( X )\) and \(( X , \varphi \circ g ^ { * } )\) , \(g \in \operatorname { A u t } ( X )\) , all have the same period point. In particular, if two marked \(K \mathcal { 3 }\) surfaces have the same period point they need not be isomorphic.
Remark 19.2.6
Remark 19.2.6. By Proposition 19.1.3, the K¨ahler cone is preserved by an isometry \(\gamma\) if and only if \(\gamma _ { \mathbb { C } }\) preserves the Hodge decomposition and maps some K¨ahler class to a K¨ahler class. This is equivalent to \({ } ^ { " } \gamma _ { \mathbb { C } }\) preserves the Hodge decomposition and the effective roots”.
Definition / construction / notation
Let us draw a further consequence of Theorem 19.2.1 which addresses general behaviour . We say that a property holds generally on some complex variety if it holds on the complement of countably many proper subvarieties. In this sense the general K3 surface is not algebraic, for instance the Kummer surface constructed from a non-algebraic complex two-torus is not algebraic. Theorem 19.2.1 implies that more is true:
Corollary 19.2.7
Corollary 19.2.7. The general \(K \mathcal { 3 }\) surface does not have any curves on it, in other words, its Picard number, i.e., the rank of its N´eron–Severi lattice, is zero.
Definition / construction / notation
For a marked K3 surface, algebraicity is equivalent to the existence of a divisor class of positive square; polarized period domains are formed by fixing such a class.
Definition / construction
The arithmetic quotient of the K3 period domain by the relevant orthogonal group is the moduli space of the corresponding lattice-polarized K3 surfaces.
Theorem 19.2.8
Theorem 19.2.8 (cf. [15, Ch. VIII, Thm 22.2]). Fix \(\ell \in \Lambda\) with \({ \boldsymbol { \ell } } \cdot { \boldsymbol { \ell } } = 2 k\) . The moduli space of \(K \mathcal { 3 }\) surfaces admitting a degree \(2 k\) polarization is the quasiprojective variety \[ { \mathsf { O } } ( \ell ^ { \perp } ) \backslash \stackrel { \circ } { D } ( \ell ^ { \perp } ) . \]
Definition 19.2.9
Definition 19.2.9. Suppose that \(S\) is a non-degenerate lattice which is either negative definite or has signature \(( 1 , r _ { - } )\) , \(r _ { - } \ge 0\) , and which is primitively embedded in \(\Lambda\) . A marking \(\varphi : H ^ { 2 } ( X , \mathbb { Z } ) \stackrel { \sim } { \to } \Lambda\) such that \(S \subset \varphi ( { \mathsf { N S } } ( X ) )\) is called an \(S\) -marking. The pair \(( X , \varphi )\) with \(\varphi\) an \(S\) -marking is called an \(S\) -marked \(\mathbf { \Delta } \mathbf { K } \mathbf { \mathcal { 3 } }\) surface. 3
Definition / construction
For a primitively embedded lattice \(S\subset\Lambda\), the period domain \(D(S^\perp)=\{[u]\in D(\Lambda):u\cdot S=0\}\) parametrizes \(S\)-marked K3 surfaces; it has dimension \(20-\operatorname{rank}S\), and outside a countable hyperplane arrangement the Néron–Severi lattice is exactly the marked copy of \(S\).
Proposition 19.2.10
Proposition 19.2.10 ([158, §2]). 1. Every non-degenerate negative definite quadratic lattice \(S\) of rank \(\rho \leq 1 1\) or of signature \(( 1 , \rho - 1 )\) , \(1 \leq \rho \leq 1 0\) , occurs as the N´eron–Severi lattice of a general \(S\) -marked \(K \mathcal { 3 }\) surface and embeds uniquely in the K3 lattice. 2. Every non-degenerate quadratic lattice \(T\) of signature \(( 2 , 2 0 - \rho )\) , \(1 2 \leq \rho \leq 2 0\) , occurs as the transcendental lattice of a general \(T ^ { \perp }\) -marked K3 surface \(^ { \cdot 4 }\) and embeds uniquely in the K3 lattice.
Proposition 19.2.11
Proposition 19.2.11 ([158, Cor. 2.9, Cor. 2.10]). Let \(X\) be a K3 surface. 1. If \(1 2 \leq \rho ( X ) \leq 2 0\) , then the N´eron–Severi lattice of \(X\) is unique in its genus. 2. If \(\rho ( X ) \leq 1 0\) , then the transcendental lattice of \(X\) is unique in its genus.
Remark 19.2.12
Remark 19.2.12. 1. In the non-algebraic situation \(\mathsf { N S } ( X )\) can indeed be negative definite or have a one-dimensional null-space. The last two cases occur if and only if the function field of \(X\) has transcendence degree \(a ( X ) = 0\) , respectively \(a ( X ) = 1\) . Indeed, if an isotropic \(f \in \mathsf { N S } ( X )\) exists, then \(\pm f\) is effective and one can show (cf. [124, Thm. 4.1,4.2]) that a suitable multiple defines an elliptic fibration. This implies that \(a ( X ) \geq 1\) . If \(a ( X ) = 2\) the surface \(X\) is algebraic (cf. [15, Ch. IV. Corollary 6.5]) and its N´eron–Severi lattice is not negative semi-definite. If \(\mathsf { N S } ( X )\) is negative definite, \(X\) cannot be elliptic or algebraic, and hence \(a ( X ) = 0\) . 2. By Theorem 15.2.6 any unimodular lattice \(S\) of rank \(\leq 2 0\) embeds uniquely in the K3 lattice provided \(T = S ^ { \perp }\) is indefinite. This ceases to be true if \(T\) is definite which may only occur in the non-algebraic situation. The simplest example is presented by the sublattice \(S = \mathbb { O } ^ { 3 } U\) . In the standard embedding its orthogonal complement is \(\mathbb { O } ^ { 2 } E _ { 8 } ( - 1 )\) . However, as we have seen (Example 1.5.1.2), there is another even unimodular rank 16 lattice which is negative definite and indecomposable, \(\Gamma _ { 1 6 } ( - 1 )\) . By Theorem 2.4.1, which gives the classification of indefinite even unimodular lattices, \(\Phi ^ { 3 } U \oplus \oplus ^ { 2 } E _ { 8 } ( - 1 ) \simeq \oplus ^ { 3 } U \oplus \Gamma _ { 1 6 } ( - 1 )\) . So two possible period domains then occur for (non-algebraic) K3 surfaces having the same N´eron–Severi lattice, but with different transcendental lattices.
Theorem 19.3.1
Theorem 19.3.1. There exists a smooth (non-Haussdorff ) analytic space M of dimension 20 together with a universal family of marked K3 surfaces. 1. The induced period map \(p : \mathcal { M } D ( \Lambda )\) is surjective; 2. The fiber over each point \([ u ] \in D ( \Lambda )\) is discrete (but might be infinite); 3. The period map \(\mathcal { n }\) is ´etale in the sense that every \([ u ] \in D ( \Lambda )\) has an open neighbourhood \(U\) so that \(\mathcal { n }\) maps every connected component of \(n ^ { - 1 } U\) biholomorphically to \(U\) .
Definition / construction / notation
The above Hodge structures for varying \([ u ] \in D ( \Lambda )\) define a variation of Hodge structure over \(D ( \Lambda )\) in the technical sense of [89] which is also called the tautological variation of Hodge structure. There does not exist a family of marked K3 surfaces \(( X _ { u } , \varphi _ { u } )\) , \(u \in D ( \Lambda )\) , with the property that the Hodge decompositions of \(H ^ { 2 } ( X _ { u } )\) under the marking \(\varphi _ { u }\) give the tautological variation, but we can achieve this locally as follows. Let \(X _ { 0 }\) be a K3-surface. There is a locally universal deformation \(p : \mathcal { X } _ { U } \to U\) of \(X _ { 0 } = p ^ { - 1 } ( 0 )\) consisting of K3 surfaces, the so-called Kuranishi family for \(X _ { 0 }\) . One may take for \(U\) a disc centered at 0 and then there is a constant marking \(\varphi _ { s } : H ^ { 2 } ( X _ { s } , \mathbb { Z } ) \stackrel { \sim } { \to } \Lambda\) over \(U\) , i.e., a trivialization of the local system \(R ^ { 2 } p _ { * } \mathbb { Z }\) . There are no two isomorphic (marked) fibres in the Kuranishi family. If this would be the case, there is an isomorphism \(g : ( X _ { 1 } , \varphi _ { 1 } ) ( X _ { 2 } , \varphi _ { 2 } )\) , and then \(\varphi _ { 1 } g ^ { * } \varphi _ { 2 } ^ { - 1 } = \mathrm { i d } _ { \Lambda }\) since \(X _ { 1 } , X _ { 2 }\) are in a single trivialized family with discrete fibres in the local system \(R ^ { 2 } p _ { * } \mathbb { Z }\) . This implies \(\varphi _ { 1 } g ^ { * } = \varphi _ { 2 }\) so that we would get identical period points by (a variation of) Lemma 19.2.5, which contradicts that the period map is an embedding (cf. [87, Part II, §2]).
Definition / construction / notation
In this alternative description, the fiber of \(\mathcal { \boldsymbol { n } }\) over a period point \(\lfloor u \rfloor\) of \(( X , \varphi )\) is constructed in such a way that it indeed parametrizes the supplementary data of Weyl chambers over \([ u ]\) which together with \(\varphi\) define a strongly marked \(\mathbf { \Delta } \mathbf { K } \mathbf { \mathcal { 3 } }\) surface. Recall first (cf. Section B.4) that the intersection form induces a nondegenerate quadratic form of signature \(( 1 , 1 9 )\) on \(\Lambda _ { u , \mathbb { R } } ^ { 1 , 1 }\) , and that the light cone has two components \(\pm C _ { u }\) . The cone \(C _ { u }\) is the union of Weyl chambers \({ \mathsf { C } } _ { u } ^ { P }\) parametrized by partitions labeled by \(P\) \[ \Delta _ { u } : = \Delta _ { u } ^ { P } \cup - \Delta _ { u } ^ { P } \]
Definition / construction
The strong-marking cover \(\widetilde D(\Lambda)\) is obtained by adjoining to each period point \(u\) a choice of signed Weyl chamber determined by a partition of the roots orthogonal to \(u\), and topologizing these local sheets by the stated Weyl-chamber gluing rule; its projection to \(D(\Lambda)\) is locally a homeomorphism.
Convention / notation
For a period point \([u]\), \(NS(u)=\{x\in\Lambda:x\cdot u=0\}\) denotes its Néron–Severi lattice.
Proposition 19.3.2
Proposition 19.3.2. There is an open dense subset \(\mathring { D } \left( S ^ { \perp } \right) \subset D ( S ^ { \perp } )\) so that the roots in \(\Delta _ { u }\) , \(\vert u \vert \in \mathcal { M } _ { S }\) , are irreducible, i.e., for some marking all roots correspond to nodal classes. Let \(P\) be a partition of the roots of \(S\) . The connected components of \(\mathcal { M } _ { S }\) consist of the non-Hausdorff smooth analytic spaces \(\mathcal { M } _ { S } ^ { \pm , P }\) of dimension \(2 0 - \operatorname { r a n k } ( S )\) .
Definition / construction / notation
Let \(I\) be an isotropic subspace of \(\Lambda _ { 2 } \otimes \mathbb { R }\) . It corresponds to a boundary component \(F _ { I } \subset \hat { D } ( \Lambda _ { 2 } )\) of \(D ( \Lambda _ { 2 } )\) whose closure is of the form \(\overline { { F } } _ { I } = \mathbb { P } ( I _ { \mathbb { C } } ) \cap \overline { { D ( \Lambda _ { 2 } ) } }\) (in the case the dimension of \(I\) is 1 this is just a point). For the SBB compactification in our setting, only the so-called rational boundary components play a role. These are the boundary components that correspond to isotropic subspaces defined over the rationals, so that we can restrict our attention to primitive isotropic sublattices \(I\) of \(\Lambda _ { 2 }\) . Since the signature of \(\Lambda _ { 2 }\) is \(( 2 , 1 9 )\) , such isotropic sublattices can only have rank 1 or 2. In the first case the boundary component is a single point, in the second case \(F _ { I }\) is isomorphic to a half-plane in \(\mathbb { P } ( I _ { \mathbb { C } } )\) . If \(I \subset J\) , where \(I\) is a rank 1 isotropic sublattice and \(J\) a rank 2 isotropic sublattice, then \(F _ { I }\) is contained in the closure of \(F _ { J }\) .
Lemma 19.4.1
Lemma 19.4.1. There is only one rank 1 primitive isotropic sublattice of \(\Lambda _ { 2 }\) up to \(\Gamma _ { 2 }\) -equivalence.
Lemma 19.4.2
Lemma 19.4.2. Let \(h = e ^ { \prime \prime } + f ^ { \prime \prime } \in \Lambda\) and \(e ^ { \prime }\) be as above. The natural maps \(0 ^ { - } ( \Lambda ) _ { h } \to 0 ( \Lambda _ { 2 } )\) and \(0 ( \Lambda _ { 2 } ) _ { e ^ { \prime } } \to 0 ( I ^ { \perp } / I )\) are surjective.
Definition / construction / notation
In our situation, the lattice is \(I ^ { \perp } / I \cong E _ { 8 } ( - 1 ) \oplus E _ { 8 } ( - 1 ) \oplus U \oplus \langle v \rangle\) , has rank 19 and is of signature \(( 1 , 1 8 )\) . We apply Vinberg’s algorithm, where we start by taking \(x = e + f \in U\) in the positive cone. As for notation, we let \(\omega _ { 1 } , \ldots , \omega _ { 8 }\) be the dual basis of \(\alpha _ { 1 } , \ldots , \alpha _ { 8 }\) and \(\omega _ { 1 } ^ { \prime } , \ldots , \omega _ { 8 } ^ { \prime }\) the dual basis of \(\alpha _ { 1 } ^ { \prime } , \ldots , \alpha _ { 8 } ^ { \prime }\) . Vinberg’s procedure leads to the following vectors (see also the Dynkin diagram for an overview):
Definition / construction
A K3 surface in arbitrary characteristic is a smooth proper surface with trivial canonical bundle and \(H^1(O_X)=0\).
Definition / construction
For a surface in positive characteristic, the \(\ell\)-adic Betti numbers are the dimensions of its \(\ell\)-adic étale cohomology groups.
Definition / construction / notation
As demonstrated in [23], the Betti numbers of a K3 surface turn out to be the same as in the complex case: \(b _ { 1 } ( X ) = 0\) and \(b _ { 2 } ( X ) = 2 2\) . However, the N´eron– Severi group can have rank \(\rho ( X )\) up to 22. We concentrate on the maximal value. A K3-surface with \(\rho ( X ) = 2 2\) is called a supersingular K3 surface. So for those surfaces \(\mathsf { N S } ( X )\) is an integral lattice of rank 22. It has signature \(( 1 , 2 1 )\) but it turns out to be non-unimodular.
Definition / construction / notation
19.5.B Supersingular K3-lattices. The N´eron–Severi lattices for supersingular K3 surfaces all belong to the class of the so-called supersingular K3-lattices defined below (cf. Definition 19.5.1).7 We explain how these can be classified using the theory we have developed. We also give their construction as given in loc. cit.
Definition 19.5.1
Definition 19.5.1. Let \(p\) be a prime number. A non-degenerate quadratic lattice \(N\) of rank \(n\) is called a supersingular \(\kappa \mathrm { 3 }\) -lattice with Artin invariant \(\boldsymbol { \mathsf { O } }\) if 1. \(n \equiv 6\) mod 8, 2. \(N\) is Lorentzian, that is, its signature is \(( 1 , n - 1 )\) , 3. \(N\) is \(p\) -elementary, i.e., \(p N ^ { * } \subset N \subset N ^ { * }\) , 4. \(N ^ { * } / N\) is an \(\mathbb { F } _ { p }\) -vector space of even dimension \(2 \sigma\) .
Construction / comparison
For a supersingular K3-lattice, the rank/signature conditions force index \(4\) modulo \(8\); at \(p=2\), type I and type II are distinguished by whether the discriminant quadratic form is \(\frac12\mathbf Z/\mathbf Z\)-valued or assumes a quarter-integral value.
Proposition 19.5.2
Proposition 19.5.2. For any odd prime \(p\) a supersingular \(K \mathcal { 3 }\) -lattice \(N\) is uniquely determined by \(n\) and \(\boldsymbol { \mathsf { O } }\) with \(n , \sigma\) as in Definition 19.5.1. Such lattices only exist if \(0 < \sigma < { \textstyle { \frac { 1 } { 2 } } n }\) . If \(p = 2\) we have \(0 < \sigma \leq { \textstyle { \frac { 1 } { 2 } } } n\) . There are two types of discriminant forms: \[ \begin{array} { r l } & { \boldsymbol { q } ^ { \# } = \mathbb { O } ^ { \sigma - 1 } \boldsymbol { u } _ { 1 } \oplus \boldsymbol { v } _ { 1 } , 1 \le \sigma < \displaystyle \frac { 1 } { 2 } \boldsymbol { n } , \quad ( T y p e ~ I ) , } \\ & { \boldsymbol { q } ^ { \# } = \langle 2 ^ { - 1 } \rangle \oplus \langle 3 \cdot 2 ^ { - 1 } \rangle \oplus \mathbb { O } ^ { \sigma - 2 } \boldsymbol { u } _ { 1 } \oplus \boldsymbol { v } _ { 1 } , \sigma \ge 2 , \quad ( T y p e ~ I I ) . } \end{array} \] If \(\sigma = 1\) one has \(q ^ { \# } = v _ { 1 }\) and if \(\sigma \ = \ { \textstyle { \frac { 1 } { 2 } } } n\) , \(q ^ { \# }\) is of type II. In all these cases there exists an up to isometry unique 2-elementary supersingular K3-lattice with discriminant form \(q ^ { \# }\) .
Remark 19.5.3
Remark 19.5.3. The above table shows that the unique supersingular lattice of rank \(n = 2 2\) and with \(\sigma = 1\) is isometric to \(U \odot N ^ { \prime }\) where \(N ^ { \prime } = D _ { 2 0 } ( - 1 )\) and so this is isometric to the Borcherds lattice from Example 17.2.10.5.
Corollary 19.5.4
Corollary 19.5.4. Let \(k\) be an algebraically closed field of odd characteristic \(p\) and let \(X\) be a supersingular K3 surface. Then NS(??) is a \(p\) -elementary \(K \mathcal { 3 }\) -lattice of rank 22 and so it is uniquely determined by the Artin invariant.
Theorem 19.5.5
Theorem 19.5.5. Let ?? have characteristic 2 and let ?? be a supersingular K3 surface over \(k\) . Then NS(??) is a supersingular K3-lattice of rank 22 of type I and so it is uniquely determined by the Artin invariant.
Theorem ([199, §3,4])
Theorem ([199, §3,4]). Supersingular \(K \mathcal { 3 }\) surfaces in characteristic 2 with Artin invariant \(\sigma\) have a moduli variety \(M _ { \sigma }\) of dimension \(\sigma - 1\) . In other words, the points of \(M _ { \sigma }\) are in one to one correspondence with isomorphism classes of supersingular \(K \mathcal { 3 }\) surfaces with Artin invariant \(\sigma\) .
Example / comparison
In characteristic \(2\), the displayed family with Artin invariant \(10\) has \(16\) raw polynomial parameters; after the indicated coordinate changes and projective reparametrizations, the effective parameter count is \(9=\sigma-1\).
Theorem ([176])
Theorem ([176]). Let \(p \neq 2 , 3\) be a prime and \(k\) an algebraically closed field of characteristic \(p\) . Supersingular K3 surfaces with Artin invariant \(\boldsymbol { \mathsf { O } }\) have a “moduli space” of dimension \(\sigma - 1\) .