FC11 — Peters–Sterk, Symmetric and Quadratic Forms (June 2024)
FC11 — Peters & Sterk, Symmetric and Quadratic Forms.
This chapter contains 14 definitions, 2 theorems, 3 propositions, 5 lemmas, 6 corollaries, 1 example, 2 constructions and 5 remarks.
Nobody has checked this chapter against Lean yet, so the statements below say nothing about whether they are formalized.
Definition / construction / notation
21.1.A Moduli for Enriques surfaces. The fixed point free involution \(j\) on the K3 surface \(X\) induces an involution \(j ^ { * }\) on \(H ^ { 2 } ( X , \mathbb { Z } )\) which can be described abstractly, using a properly chosen isometry \(H ^ { 2 } ( X , \mathbb { Z } ) \ \simeq \Lambda\) . To explain this, we rewrite the K3 lattice as \(\Lambda = U \oplus W \oplus W ^ { \prime }\) , where \(W , W ^ { \prime }\) are copies of the Enriques lattice. This last decomposition can be used to define the lattice Enriques involution \[ \iota _ { \mathsf { E n r } } ( u , v , v ^ { \prime } ) = ( - u , v ^ { \prime } , v ) , \quad ( u , v , v ^ { \prime } ) \in U \oplus W \oplus W ^ { \prime } . \]
Definition / construction / notation
Such a marking \(\varphi\) is called a marking for \(( X , j )\) . Since \(Y = X / G = X / \langle j \rangle\) has no non-zero holomorphic 2-forms, the involution \(j ^ { * }\) acts as −id on \(H ^ { 2 , 0 } ( X )\) and hence also on the corresponding period point \([ \omega ] = \varphi [ H ^ { 2 , 0 } ( X ) ]\) . So \([ \omega ]\) belongs to \(D ( \Lambda ) \cap \mathbb { P } ( \Lambda _ { G } ) = D ( \Lambda _ { G } )\) where \(\Lambda _ { G }\) is the anti-invariant lattice. For later reference, we observe \[ \Lambda _ { { \cal G } } = \{ ( u , w , - w ) \in { \cal U } \oplus { \cal W } \oplus { \cal W } ^ { \prime } \} \simeq { \cal U } \oplus \Lambda _ { \mathsf { E n r } } ( 2 ) . \]
Definition / construction / notation
and that, conversely, such points correspond to period points for Enriques surfaces coming from marked pairs \(( X , j )\) of a K3 surface equipped with an Enriques involution. Such period points will be called period points of Enriques surfaces. The group \[ \Gamma _ { \mathsf { E n r } } : = \{ \gamma \in \mathsf { O } ( \Lambda ) \mid \gamma \circ l _ { \mathsf { E n r } } = \iota _ { \mathsf { E n r } ^ { \circ } } \gamma \} \]
Definition 21.1.1
Definition 21.1.1. For any subset \(R\) of a lattice, we let \(\langle R \rangle\) be the smallest primitive sublattice containing \(R\) . 1. An ordered pair of \(\left( - 2 \right)\) -roots \(\{ r , \iota _ { \mathsf { E n r } } ( r ) \}\) , \(r \in \Lambda\) , which satisfy \(r \cdot \iota _ { \mathsf { E n r } } ( r ) = 0\) is called an \(\iota _ { \mathsf { E } \mathsf { n r } }\) -adapted root pair . The collection of all \(\iota _ { \mathsf { E } \mathsf { n r } }\) -adapted root pairs of \(\Lambda\) is denoted by \(\Delta ( \Lambda , \iota _ { \mathsf { E n r } } )\) . If \(\{ r , \iota _ { \mathsf { E n r } } ( r ) \} \in \Delta ( \Lambda , \iota _ { \mathsf { E n r } } )\) , we set \(r ^ { \pm } : = r \pm \iota _ { \mathsf { E n r } } ( r )\) . The corresponding root \(\pmb { r } \in \Lambda _ { \mathsf { E n r } }\) in the Enriques lattice is given by \(\pi ^ { * } r = r ^ { + }\) . In particular \(\Delta ^ { \pm } ( X ) ) \subset { \mathsf { N S } } ( X )\) . 2. Two sets \(R , R ^ { \prime }\) of roots in \(\Lambda\) are said to be equivalent if the sublattices \(\langle R \rangle\) and \(\langle R ^ { \prime } \rangle\) of \(\Lambda\) are isometric under \({ \mathrm { O } } ( \Lambda )\) . 3. Geometry enters by passing to roots that are nodal classes, that is, classes of smooth \(\left( - 2 \right)\) -curves: \[ \begin{array} { r c l } { { \Delta ( X ) } } & { { : = } } & { { \{ \{ r , \iota _ { \mathsf { E n r } } ( r ) \} \in \Delta ( \Lambda , \iota _ { \mathsf { E n r } } ) \mid r \mathrm { ~ i s ~ a ~ n o d a l ~ c l a s s } \} , } } \\ { { \Delta ^ { \pm } ( X ) } } & { { : = } } & { { \{ r ^ { \pm } \mid \{ r , \iota _ { \mathsf { E n r } } ( r ) \} \in \Delta ( X ) \} , } } \\ { { \Delta ( Y ) } } & { { : = } } & { { \{ r \mid \{ r , \iota _ { \mathsf { E n r } } ( r ) \} \in \Delta ( X ) \} \subset \Lambda _ { \mathsf { E n r } } . } } \end{array} \]
Construction / comparison
For an Enriques-adapted root pair, one member is a nodal class exactly when its image under the Enriques involution is a nodal class.
Definition / construction / notation
In this situation we say that \(( X , j )\) is of nodal type \(\Delta ( X )\) .
Proposition 21.1.2
Proposition 21.1.2. If \(\{ r , \iota _ { \mathsf { E n r } } ( r ) \}\) and \(\{ s , \iota _ { \mathsf { E n r } } ( s ) \}\) are two \(\iota _ { \mathsf { E } \mathsf { n } \mathsf { r } }\) -adapted root pairs, there exists \(\gamma \in \Gamma _ { \mathsf { E n r } }\) for which \(\gamma ( r ^ { - } ) = s ^ { - }\) .
Remark: Remark
Remark. The above result implies that after a possible change of marking, we may assume that for any given \(\iota _ { \mathsf { E } \mathsf { n } \mathsf { r } }\) -adapted root pair \(\{ r , \iota _ { \mathsf { E n r } } ( r ) \}\) , the associated root \(r ^ { - }\) is of the form \(\rho - \iota _ { \mathsf { E n r } } ( \rho )\) where \(\rho\) is any root contained in the \(W\) -summand. Indeed, for such a choice, \(\{ \rho , \iota _ { \mathsf { E n r } } ( \rho ) \}\) is an \(\iota _ { \mathsf { E } \mathsf { n } \mathsf { r } }\) -adapted root pair such that \(\rho ^ { - }\) is in the \(\operatorname { O } ( \Lambda _ { G } )\) -orbit of \(r ^ { - }\) .
Proposition 21.1.3
Proposition 21.1.3. Let \(( X , j )\) be a pair of a \(K \mathcal { 3 }\) surface equipped with an Enriques involution and let \(Y = X / G\) be the quotient Enriques surface. Use a marking for \(( X , j )\) to identify \(H ^ { 2 } ( X )\) with \(\Lambda\) and \(H ^ { 2 } ( Y ) /\) Tors with \(\Lambda _ { \mathsf { E n r } } = \Lambda ^ { G } \big ( \frac { 1 } { 2 } \big )\) . Let \(\Delta ( X )\) be the nodal type of \(X\) and set \[ \begin{array} { r l } & { \mathsf { C } _ { X , j } ^ { \mathrm { a m p } } : = \{ x \in \Lambda ^ { G } \otimes \mathbb { R } \mid x \in \mathsf { C } _ { X } , x \cdot r ^ { + } > 0 \ f o r \ a l l \ \{ r , \iota _ { \mathsf { E n r } } ( r ) \} \in \Delta ( X ) \} } \\ & { \quad \quad \quad = \{ y \in \Lambda _ { \mathsf { E n r } } \otimes \mathbb { R } \mid y \in \mathsf { C } _ { Y } , y \cdot r > 0 \ f o r \ a l l \ r \in \Delta ( Y ) \} . } \end{array} \] Then \(\mathsf { C } _ { X , j } ^ { \mathrm { a m p } } = \mathsf { C } _ { X } ^ { \mathrm { a m p } } \cap \Lambda ^ { G } \otimes \mathbb { R }\) corresponds to the ample cone of \(Y\) . Its closure is a fundamental domain for the action of \(\mathsf { W } ^ { - } ( \Delta ^ { + } ( X ) ) : = s u b g r o u p\) of \({ \mathsf { O } } ( \Lambda ^ { G } )\) generated by \(\sigma _ { r ^ { \circ } } \sigma _ { \iota _ { \mathsf { E n r } } } ( r ) = \sigma _ { r ^ { + } }\) , \(\{ r , \iota _ { \mathsf { E n r } } ( r ) \} \in \Delta ( X )\) on the positive cone of \(\Lambda ^ { G } \otimes \mathbb { R }\) . The latter Weyl group depends only on the set of nodal classes on \(Y\) , more precisely, on \(\Delta ( Y )\) .
Corollary 21.1.4
Corollary 21.1.4. As in the preceding paragraph, assume that \(\gamma \in \Gamma _ { \mathsf { E n r } }\) preserves the period point of an Enriques surface \(X / \langle j \rangle\) (and hence induces a Hodge isometry on \(H ^ { 2 } ( X , \mathbb { Z } ) ,\) ). Then for some \(w \in \mathsf { W } ^ { - } ( \Delta ^ { + } ( X ) )\) the isometry ??◦?? is induced by \(a\) unique automorphism of \(X\) commuting with \(j\) .
Lemma 21.1.5
Lemma 21.1.5. Let ?? be the period point of a marked pair \(( X , j )\) , \(X\) a \(K \mathcal { 3 }\) surface \(X\) equipped with an Enriques involution \(j\) , and let the nodal set \(\Delta _ { \omega }\) be given by equation (21.6). Then any partition \(\Delta _ { \omega } = \Delta _ { \omega } ^ { + } \cup - \Delta _ { \omega } ^ { + }\) closed under taking sums determines \(\Delta ( X )\) up to equivalence. In other words, the nodal type of \(X\) is completely determined by \(\Delta _ { \omega }\) .
Remark 21.1.6
Remark 21.1.6. Let \(\omega\) be the period point of a \(j\) -marked K3 surface \(( X , \varphi )\) and let \(\gamma \in \Gamma _ { \mathsf { E n r } }\) . Then \(\omega ^ { \prime } = \gamma ( \omega )\) is the period point of an isomorphic \(j ^ { \prime }\) -marked K3- surface \(( X ^ { \prime } , \gamma \circ \varphi )\) . Under ?? the sets \(\Delta _ { \omega }\) and \(\Delta _ { \omega ^ { \prime } }\) correspond and the same holds for their Weyl chambers. If \(r \in \Delta ( X )\) is indecomposable, also \(\gamma ( r )\) is indecomposable. Hence isomorphic Enriques surfaces have the same nodal type.
Definition / construction / notation
For an Enriques surface of nodal type \(\Delta(X)\), the \((+)\)-root invariant is \(\delta^+(X)=r_2(\Delta^+(X))\subset\Lambda^G\otimes\mathbf F_2\), where \(r_2\) is reduction modulo \(2\).
Lemma 21.1.7
Lemma 21.1.7. Let \(H : = \ker \xi \subset \langle \Delta ^ { - } ( X ) \rangle \otimes \mathbb { F } _ { 2 }\) . Then the map \(\xi\) establishes an isomorphism \([ \langle \Delta ^ { - } ( X ) \rangle \otimes \mathbb { F } _ { 2 } ] \ / H \stackrel { \simeq } { \to } \delta ^ { + } ( X )\) .
Definition / construction
The pair \((\langle\Delta^-(X)\rangle\otimes\mathbf F_2,H)\), where \(H=\ker\xi\), is the negative root invariant; equivalent positive root invariants induce equivalent negative root invariants.
Lemma 21.1.8
Lemma 21.1.8. \(\langle \Delta ^ { - } ( X ) \rangle ( \Sigma )\) is a negative definite root lattice (and hence isometric to an orthogonal direct sum of the negative definite root lattices of types ??-??- \(E\) ). The rank of \(\langle \Delta ^ { - } ( X ) \rangle\) is at most 10 and so \(\langle \Delta ^ { - } ( X ) \rangle\) belongs to finitely many isometry types.
Corollary 21.1.9
Corollary 21.1.9. There are finitely many possible nodal invariants \(\Delta ( X )\) (up to equivalence) and hence an Enriques surface can have at most finitely many nodal types.
Lemma 21.1.10
Lemma 21.1.10. Let \(( X , j )\) be a \(K \mathcal { 3 }\) -surface equipped with a fixed point free involution \(j\) , and with nodal type \(\Delta ( X )\) . A marking for \(( X , j )\) identifies the transcendental lattice \(T\) of \(X\) with a sublattice of \(\langle \Delta ^ { - } ( X ) \rangle ^ { \bot } \cap \Lambda _ { G }\) .
Definition / construction / notation
Observe now that for any \(x \in \Lambda\) the linear forms \(b ( x , - )\) and \(b ( \iota _ { \mathsf { E n r } } ( x ) , - )\) evaluated on \(\Lambda _ { G }\) only differ by a sign and so the hyperplanes \(H _ { x } = \{ [ \omega ] \in D _ { \Lambda _ { G } } |\) \(\boldsymbol { \omega } \cdot \boldsymbol { x } = 0 \}\) and \(H _ { \iota _ { \mathsf { E n r } } ( x ) }\) are the same. This applies in particular to \(\iota _ { \mathsf { E } \mathsf { n } \mathsf { r } }\) -adapted root pairs \(\{ r , \iota _ { \mathsf { E n r } } r \}\) so that the hyperplane \(D _ { \mathsf { E n r } } \cap H _ { r }\) in fact only depends on \(r ^ { - }\) . For a collection \(\Delta\) of \(\iota _ { \mathsf { E } \mathsf { n r } }\) -adapted root pairs, the intersection of the hyperplanes orthogonal to the roots thus only depends on \(\Delta ^ { - }\) and we denote it by \[ \begin{array} { r } { D _ { \mathsf { E n r } } ^ { \Delta ^ { - } } = \bigcap _ { r \in \Delta } D _ { \mathsf { E n r } } \cap H _ { r } = \{ [ \omega ] \in D _ { \mathsf { E n r } } \mid \Delta _ { \omega } = \Delta \} , } \end{array} \]
Example / comparison
Example 21.1.11. For a general nodal Enriques surface, the moduli space has dimension \(9\). With \(S\) the primitive lattice generated by \(\Lambda^G\) and the nodal root and \(T=S^\perp\), one has \(T\simeq U\oplus U(2)\oplus E_7(-2)\), \(q_S^\#=-q_T^\#\), and \(S\simeq U\oplus\langle-4\rangle\oplus E_8(-2)\).
Definition / construction
The mod-\(2\) congruence subgroup is the kernel of the reduction action on the mod-\(2\) lattice/discriminant data.
Lemma 21.2.1
Lemma 21.2.1. The map \(\rho _ { 2 }\) is surjective and \(\mathrm { k e r } ( \rho _ { 2 } ) = { \sf O } ^ { \# } ( \Lambda ^ { G } ) = { \sf O } ( \Lambda _ { \sf E n r } ) [ \frac { c } { 1 - \lambda _ { \sf } }\) 2].
Definition / construction / notation
The following concept similar to the terminology for K3 surfaces was introduced by S. Mukai and H. Ohashi in [161]:
Definition 21.2.2
Definition 21.2.2. As before, let \(Y = X / G\) be an Enriques surface. An automorphism \(g \in \operatorname { A u t } ( Y )\) is semi-symplectic if one of its two lifts to \(X\) is symplectic. The group of semi-symplectic transformations of \(Y\) is denoted \(\operatorname { A u t } _ { s } ( Y )\) .
Construction / comparison
For a general non-nodal Enriques surface, the K3 cover has Picard number \(10\); at a \(K_{10}\)-generic period point the genericity result from Chapter 20 forces all automorphisms to be semisymplectic.
Theorem 21.2.3
Theorem 21.2.3 ([16, 172]). Let \(Y\) be an Enriques surface without nodal curves. Then \[ \mathrm { A u t } _ { s } ( Y ) \simeq { \sf O } ^ { \# , - } ( \Lambda ^ { G } ) \simeq { \sf O } ^ { - } ( \Lambda _ { \sf E n r } ) [ 2 ] . \] In particular, Enriques surfaces without nodal curves have an infinite group of automorphisms. If the period point of \(Y\) in \(D ( \Lambda _ { G } )\) is \(K _ { 1 0 }\) -generic, then all automorphisms of \(Y\) are semi-symplectic.
Corollary 21.2.4
Corollary 21.2.4. A non-nodal \(K _ { 1 0 }\) -generic Enriques surface ?? has 17·31 distinct elliptic fibrations.
Definition / construction / notation
For an Enriques surface of fixed nodal type \(\Delta=\Delta(X)\), write \(\Delta^\pm=\Delta^\pm(X)\) and identify the nodal set of the quotient \(Y=X/G\) with \(\Delta\).
Definition / construction
For fixed Enriques nodal type \(\Delta\), let \(S=\langle\Lambda^G,\Delta^-\rangle\) be the smallest primitive sublattice of the K3 lattice generated by the invariant lattice and \(\Delta^-\).
Definition / construction
The period domain of Enriques surfaces of nodal type \(\Delta\) is \(D_{\mathrm{Enr}}(S)=D(S)\cap D_{\mathrm{Enr}}\subset D(S)\cap D(\Lambda_G)\).
Theorem 21.2.5
Theorem 21.2.5. Let \(Y\) be a general Enriques surface of nodal type \(\Delta\) . Its group of semi-symplectic automorphisms is given by \(\mathrm { A u t } _ { s } ( Y ) = G ( \Delta ^ { - } ) / W ^ { - } ( \langle \Delta ^ { + } \rangle )\) .
Remark 21.2.6
Remark 21.2.6. The group \(\mathsf { O } ^ { - } ( \langle \Delta ^ { - } \rangle )\) contains \({ \mathsf { W } } ^ { - } ( \Delta ^ { - } )\) . By Lemma 21.1.8 \(\Delta ^ { - } ( - \textstyle { \frac { 1 } { 2 } } )\) is a root lattice and so is an orthogonal sum of (classical) irreducible root lattices of type \(A\) -??- \(E\) . This decomposition is unique up to permuting isometric irreducible sublattices (by the unicity of the decomposition of a positive definite lattice stated by Theorem 1.12.3) and so by Corollary 17.2.3, \(0 ^ { - , \# } ( \langle \Delta ^ { - } \rangle ) / \mathsf { W } ^ { - } ( \Delta ^ { - } )\) is a product of permutation groups coming from isometric factors of the same type provided \(\Delta ^ { - } ( - \textstyle { \frac { 1 } { 2 } } )\) does not contain irreducible sublattices of type \(D _ { 4 }\) . In that case the occurrence of such a factor contributes an extra factor, namely the cyclic group \(C _ { 3 }\) .
Proposition 21.2.7
Proposition 21.2.7. Let \(N\) be the sublattice of the Enriques lattice generated by the set \(\pmb R\) of roots \(\pmb { s } \in \Lambda _ { \mathsf { E n r } }\) corresponding to \(\iota _ { \mathsf { E } \mathsf { n } \mathsf { r } }\) -adapted root pair such that \(s ^ { - } = r ^ { - }\) . Then \(N = \mathbb { Z } r \oplus 2 \cdot r ^ { \perp }\) and the roots of \(N\) are precisely the roots that make up \(\Delta ( Y )\) .
Remark 21.2.8
Remark 21.2.8. Using a reduction to \(\left( - 4 \right)\) -roots whose \(U\) -component is equal to \(0\) , one can show that \(\Gamma _ { \mathsf { E n r } }\) acts transitively on roots \(s \in \Delta ( X )\) for which \(s ^ { - } = r ^ { - }\) .
Corollary 21.2.9
Corollary 21.2.9. The group \(\mathsf { O } ^ { - } ( N ) / \mathsf { W } ^ { - } ( N ) = \mathsf { O } ( N ) / \mathsf { W } ^ { - } ( N )\) contains an infinite abelian free group of rank 7.
Corollary 21.2.10
Corollary 21.2.10 ([173, 187]). The group of semi-symplectic automorphisms of the general nodal Enriques surface \(Y = X / \langle j \rangle\) is isomorphic to \[ \mathsf { O } ( \Lambda _ { \mathsf { E n r } } ) [ 2 ] \times \{ \mathrm { i d } , s _ { r } \} / \mathsf { W } ^ { - } ( N ) , \] where \(N \simeq \langle - 2 \rangle \oplus 2 \cdot ( U \oplus E _ { 7 } ( - 1 ) )\) . If, moreover the period point of \(( X , j )\) is \(K _ { 1 1 }\) -generic, then all automorphisms of \(Y\) are semi-symplectic.
Corollary 21.2.11
Corollary 21.2.11. The automorphism group of a general nodal Enriques surface contains a free abelian group of rank 7 and hence is infinite.
Remark 21.2.12
Remark 21.2.12. Recall (cf. Lemma 4.1.5) that \(\widetilde { T } _ { 2 , 3 , 6 }\) is an isometric copy of the Enriques lattice. Let \(r\) be the root \(\alpha _ { 9 }\) in the diagram for Lemma 4.1.5. Then \(\mathbb { Z } r \oplus r ^ { \perp } \subset \Lambda _ { \mathsf { E n r } }\) is isometric to \(\widetilde { T } _ { 2 , 4 , 6 } \subset \widetilde { T } _ { 2 , 3 , 6 }\) , the so-called Reye lattice. Note that \(N = \mathbb { Z } \cdot r \oplus 2 r ^ { \perp }\) has index 2 in the Reye lattice. Since \(\widetilde { T } _ { 2 , 4 , 6 }\) has index 2 in \(\Lambda _ { \mathsf { E n r } }\) , using the mod 2 reduction \(\rho _ { 2 } : \Lambda _ { \mathsf { E n r } } \to \Lambda _ { \mathsf { E n r } } / 2 \Lambda _ { \mathsf { E n r } }\) , one can characterize the Reye lattice as \[ \widetilde { T } _ { 2 , 4 , 6 } = \rho _ { 2 } ^ { - 1 } ( \rho _ { 2 } ( \pmb { r } ) ) = \{ x \in \Lambda _ { \mathsf { E n r } } \mid x \equiv \pmb { r } \bmod 2 \Lambda _ { \mathsf { E n r } } \} . \] while \(N = \ker \rho _ { 2 } | { \boldsymbol { r } } ^ { \perp }\) . Using this, D. Allcock [1] and F. Cossec–I. Dolgachev [45] gave an alternative proof of Corollaries 21.2.9 and 21.2.10.