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800 CONICS IN A SMOOTH QUARTIC SURFACE
ALEX DEGTYAREV
Abstract.
We construct an example of a smooth spatial quartic surface thatcontains 800 irreducible conics. Introduction
This short note was motivated by Barth, Bauer [1], Bauer [2], and my recentpaper [4]. Generalizing [2], define N n ( d ) as the maximal number of smooth rationalcurves of degree d that can lie in a smooth degree 2 n K X ⊂ P n +1 . (Allalgebraic varieties considered in this note are over C .) The bounds N n (1) have along history and currently are well known, whereas for d = 2 the only known valueis N (2) = 285 (see [4]). In the most classical case 2 n = 4 (spatial quartics), thebest known examples have 352 or 432 conics (see [1, 2]), whereas the best knownupper bound is 5016 (see [2], with a reference to S. A. Strømme).For d = 1, the extremal configurations (for various values of n ) tend to exhibitsimilar behaviour. Hence, contemplating the findings of [4], one may speculate that • it is easier to count all conics, both irreducible and reducible, and • nevertheless, in extremal configurations all conics are irreducible.On the other hand, famous Schur’s quartic (the one on which the maximum N (1)is attained) has 720 conics (mostly reducible), suggesting that 432 should be farfrom the maximum N (2). Therefore, in this note I suggest a very simple (althoughalso implicit) construction of a smooth quartic with 800 irreducible conics. Theorem 1.1 (see § . There exists a smooth quartic surface X ⊂ P containing irreducible conics. I conjecture that N (2) = 800 and, moreover, 800 is the sharp upper bound onthe total number of conics (irreducible or reducible) in a smooth spatial quartic. Acknowledgements.
I am grateful to S lawomir Rams who introduced me to thesubject and made me familiar with the current state of the art; it is his curiositythat encouraged my work in this direction.2.
The Leech lattice (see [3])2.1.
The Golay code.
The ( extended binary ) Golay code is the only binary codeof length 24, dimension 12, and minimal Hamming distance 8. We regard codewordsas subsets of Ω := { , . . . , } and denote this collection of subsets by C ; clearly, |C| = 2 . The code C is invariant under the complement o Ω r o . Apart from ∅ and Ω itself, it consists of 759 octads (codewords of length 8), 759 complementsthereof, and 2576 dodecads (codewords of length 12). Key words and phrases. K The setwise stabilizer of C in the full symmetric group S (Ω) is the Mathieu group M of order 244823040; the actions of this group on Ω and C are described in detailin § The square vectors. The
Leech lattice is the only root-free unimodulareven positive definite lattice of rank 24. For the construction, consider the standardEuclidean lattice E := L i Z e i , i ∈ Ω, and divide the form by 8, so that e i = 1 / − / appearing throughout in [3].) Then, the Leechlattice is the sublattice Λ ⊂ E spanned over Z by the square 4 vectors of the form(2.1) ( ∓ , ± ) (the upper signs are taken on a codeword o ∈ C ) . (We use the notation of [3]: a m , b n , . . . means that there are m coordinates equalto a , n coordinates equal to b , etc .) Apart from (2.1), the square 4 vectors in Λ are( ± , ) ( ± , or(2.2) ( ± , ) (no further restrictions) . (2.3)Altogether, there are 196560 square 4 vectors: 24 · |C| = 98304 vectors as in (2.1),2 ·
759 = 97152 vectors as in (2.2), and 2 · C (24 ,
2) = 1104 vectors as in (2.3).3.
The construction
In this section, we prove Theorem 1.1.3.1.
The lattice S . Consider the lattice V := Z~ + Z a + Z u + Z u + Z u withthe Gram matrix −
10 0 2 4 00 1 − . It can be shown that, up to O (Λ), there is a unique primitive isometric embedding V → Λ; however, for our example, we merely choose a particular model. Fix anordered quintuple Q := (1 , , , , ⊂ Ω and choose one of the four octads O suchthat O ∩ Q = { , , , } ( cf. sextets in § O = { , , , , , , , } (the underlined positions in the toprow of Table 1). Then, the generators of V can be chosen as shown in the upperpart of Table 1. (For better readability, we represent zeros by dots; all componentsbeyond ¯ O := Q ∪ O are zeros.)The choice of Q and O is unique up to M ; furthermore, the subgroup G ⊂ M stabilising Q pointwise and O as a set can be identified with the alternating group A ( O r Q ); in particular, it acts simply transitively on the set of ordered pairs(3.1) ( p, q ) : p, q ∈ O r Q = { , , , } , p = q. Define a conic as a square 4 vector l ∈ Λ such that l · ~ = 2 , l · a = 1 , l · u = l · u = l · u = 0 . This strange condition can be recast as follows: l · ~ = 2 and l (as well as ~ ) lies inthe rank 20 lattice S := ¯ V ⊥ ⊂ Λ , where ¯ V := ~ ⊥ V . Using § l is as in (2.3), we
00 CONICS IN A SMOOTH QUARTIC SURFACE 3
Table 1.
The lattice V and the conics ~ · · · · · · · a · · · · · · · u · · · · · · · u · · · · · · · u − · − − − − ∗ − ∗ ± − − ∗ − ∗ ± · · · · · · · ± , · · · · · ∗ − ∗ ± , fixed in Q movable in O r Q Table 2.
The number of conics in S : C (4 , ·
16 = 96 (sets o ∈ C such that o ∩ ¯ O = { , , , p, q } ) , : C (4 , ·
16 = 96 (sets o ∈ C such that o ∩ ¯ O = { , , p, q } ) , : 2 ·
10 = 320 (octads o ∈ C such that o ∩ ¯ O = { , } ) , : 2 · P (4 , · o ∈ C such that o ∩ ¯ O = { , , p, q } ) . have l · a = 0 mod 2.) The number of conics within each pattern is computed asshown in Table 2, where • the ordered or unordered pair ( p, q ) as in (3.1) designates the two variablespecial positions marked with a ∗ in Table 1, • the underlined factor counts certain codewords o ∈ C ; the restrictions givenby (2.1) or (2.2) are described in the parentheses, and • the other factors account for the choice of ( p, q ) and/or signs in ± The N´eron–Severi lattice.
Observe that ~ ∈ S ∨ : indeed, ~ − a ∈ ¯ V andwe have x · ~ = 2 x · a = 0 mod 2 for any x ∈ S . Thus, we can apply to S ∋ ~ theconstruction of [4], i.e. , consider the orthogonal complement ~ ⊥ S = V ⊥ ⊂ Λ, reversethe sign of the form, and pass to the index 2 extension N := (cid:0) − ( ~ ⊥ S ) ⊕ Z h (cid:1) ∼ , h = 4 , containing the vector c := c ( l ) := l − ~ + h for some (equivalently, any) conic l ∈ S . These new vectors c ∈ N are also called conics ; one obviously has(3.2) c = − c · h = 2 . They are in a bijection with the conics in S ; hence, there are 800 of them.Starting from discr V ∼ = (cid:20) (cid:21) ⊕ (cid:2) (cid:3) ⊕ (cid:2) (cid:3) (see Nikulin [6] for the concept of discriminant form discr V := V ∨ /V and relatedtechniques), we easily compute N := discr N ∼ = (cid:2) (cid:3) ⊕ (cid:2) (cid:3) ⊕ (cid:2) (cid:3) ∼ = (cid:2) − (cid:3) ⊕ (cid:2) − (cid:3) ⊕ (cid:2) (cid:3) . ALEX DEGTYAREV
Therefore, −N ∼ = discr T , where T := Z b ⊕ Z c , b = 4, c = 40. Then, it followsfrom [6] that there is a primitive isometric embedding of the hyperbolic lattice N to the intersection lattice H of a K T ∼ = N ⊥ play the rˆole ofthe transcendental lattice. Finally, by the surjectivity of the period map [5], weconclude that there exists a K X with NS ( X ) ∼ = N .3.3. Proof of Theorem 1.1.
The N´eron–Severi lattice NS ( X ) ∼ = N constructedin the previous section is equipped with a distinguished polarisation h ∈ N , h = 4.Since the original lattice S ⊂ Λ is root free, N does not contain any of the following“bad” vectors: • e ∈ N such that e = − e · h = 0 ( exceptional divisors ) or • e ∈ N such that e = 0 and e · h = 2 (2 -isotropic vectors )(see [4] for details). Hence, by Nikulin [7] and Saint-Donat [8], the linear system | h | is fixed point free and maps X onto a smooth quartic surface X ⊂ P .The lattice N contains 800 conics c as in (3.2). By the Riemann–Roch theorem,each class c is effective, i.e. , represented by a curve C ⊂ X of projective degree 2.Since X is smooth and contains no lines (or other curves of odd degree, as we have h ∈ N ∨ by the construction), each of these curves C is irreducible. This concludesthe proof of Theorem 1.1. (cid:3) References
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