R.eclbjdo· Octybre 1992. 

On a pencil of K3 surfaces • 

Víctor González Aguilera 

Resumen. 
Una superficie K3 es una superficie analítica compleja 

S(dimcS = 2) que es simplemente conexa y cuyo fibrado 
canónico K 5 es triv ia l. Para las superficies K3 que son 
algebraicas (es decir, inmersas en Pn), existe un espacio de 
módulos M(K3) que es 19 dimensional. 

En esta nota se construye una subvariedad !-dimensional 
de M (K3) con un grupo de simetrías fijo (Z/4'71..) 2 x G.<t) y 
se describen explícitamente sus degeneraciones. 

Let P3 be th e complex projective spac.e with coord ina.tes (x1 , x2,x3,x4) E P3. 
The section FE Hº(P3 , Op1 (4)) given by F(x 1,x2 ,x3 ,x4 ) = xf+:z4+x1+xli defines 
the classical Fermat's quartic. Consider the section TE Hº(P3 ,0P,(4)) given by 
T(x1, x2 1 x 3,x4) = x 1x 2x 3x 4, as F and Tare li nearl y independent tbe 2-dimensional 
vector subspace S = {F, T} e H º(P3 ,0p3 (4)) defines a pencil of quartic surfaces 
that we will denote by: 

80 = {F+ofl'/a E Pi} 

Each generi c quart ic S0 of the pencil is smooth, by Lefschetz's theorem on hy-
perplanesections, S0 issimply-connected, furthermore as K s,. ~ Os. (4 - (3 + 1)) ~ 
Os. the canonical sheaf is trivial and S0 is a K3 surfac.e. 

We b egin d esc.ribing the base locus of our pencil. 

Propositioo 1 Th e bas e locus o/ (S0 ) 0 ep1 denoted by B (S0 ) consi.sts o/ a. sta-
ble curve o/ genus g = 33 with a.sso ciated groph given b31 Fig J. 

Proof We co nsider tb e four hyperplanes H¡ = {x¡ =O) with i E {l , ... ,4 ). The 
intersection of F =O with each hyperplane H¡, i E {1 , ... , 4) , gives a Fermat's 
quartic of ge nus g = 3, that we denote by F¡ , i E {l , .. . , 4) . Tbe intersection F¡ n Fj 
is contained in each line F¡ n FJ = L¡; and consists of four differeots points. 

ºPa.rt ial ly 1upp«Ud by FONDECYT 0760/92 

21 



22 CUBOS V. Gonzál .. A. 

Then the base locus is a stable curve, tbat consists of four curves F';. o( genua 
g = 3, where each F¡ cuts each of the other three F; at four point.s and tbe grapb 
associated corresponds to Fig. 1, the genus g = 33 followa from the well knowo 
formula [IJ for graph curves. 8 

It is well known that the biggest group of proj ective automorphisms of F = O 
{Ferma t 's quartic surface) is isomorphic to Ge!: (7l/4Z )3 x 94 , where 9 4 is the sym· 
metric group. T he group G can be represented in PGL(4,0:::) as 9i; t (x1, x2 ,x3 , x4) = 
(~;x1 ,~;x2,ex3 ,x4) with ~a prim..itive rooth o f the unity of order 4 and Y4 acting 
by permutation of coordinates. 

As the section T E Hº(P3, Qps{4)) is also invariant by G, tben G ~ Aut(T) 
and the 2-dimensional vector S'ubspace S = (F, T ) e Hª(P3 ,0P3)) is G-invariant. 

We bave tbe following proposition. 

Propositioo 2 The group G acts on the pencil by multiplication by i and the 
subgroup H 3:: (Z / 47l.)2 x 94 acts trivially. 

Proof By definition the subspace Se Hº(P3 ,0p3 (4)) is invariant by the action 
of G the n the group G acts on the pencil. 

We consider the epimorphism: 

µ: Z/4Z X Z/4Z X Z / 4Z ~ Z /4Z 

given by µ((i,(i,(k) = eieie, ker µ is isomorphic to Z / 4Z X '11../ 4Z and each 
h E H = (Z x Z / 4Z)2 acts trivially on T , a.s g,. acts also t rivially on this section, 
we have the result. • 

Aa elementary calculatio n allows to ob tain the values of a E P1 where 80 is 
singular and we have the following proposit ion. 

Propositioo 3 For each proiective K, surface in the pencil (S0 )oeP1 we have 
tha.t H ~ Aut (Sa) · There are only five t1ingula.r fibert1 in the pencil1 they 
coM"eSpond to the values a= {i, -1 , -i, l} anda= oo. For a-:/; 00 the sin-
gular fibers are Kummer's sur/ace, for a= oo the singular fiber i3 o/ type 

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CUBO 8 23 

111, a r.a.t.ional surface meeting a.long ra.tional curves its dual gmph is the 
tetrahedron. 

Pr00f For each a E { i, -1, -i1 1} an easy calculation proves tbat S0 has the 
maximum of ordinary double points (16 double point.s} then by (2} S0 is a Kummer 
surface. 

For a = oo it i.s clear that 800 consists of the union of four rat.ional sul'faces 
{~he hyper.planes H¡ = {x, =O}, i E {1, ... 4}) where two of them intersects along a 
rational curve. • 

Remark. As every K1mrmer surface is isomorphic to a surface of the for.m 
:r a.c ~C)/{±1}, where C is a curve of genus 2, and :r ac(C) denotes tbe Jacobi vari~ 
ety. Then in our case .7 ac(C) must admit 9 4 as reduced group of automorpbisms. 

Finally it would be interesting to study bhe variation of the Picard number of 
s. as in /4]. 

References 

[l] Miranda, R., Graph Ou.rves and Curves on K3 Surfaces, Lectures on 
Riemann Suda.ces, I.C.T.P., 1987. World Scienliific. 

[2] Nikulin, V., On K u.mmer su.rfa,ces, Math. USRR Izvestija 9 1 261-275 
(1975). 

[3J SafareviC1 I.R., Algebmic surfa,ces, Proceedings of the Stek.lov Institute of 
Mathematics, No. 75 AMS 1967. 

[4] Shioda, T. On the Pica,rd number of a. complex projective variety, Annales 
Scientifiques de L'ENS, Tome 14·, fase 3 1 303-321. 1981. 

Direcci6n del autor; 
Departamento de Matemática 

Un iversidad Técnica Feder:ico Santa María. 
e.mail VGONZALE@UTFSM.BITNET 

Casilla 110-V. Valparaiso. 




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