(de Gruyter 2003
Caps on Hermitian varieties and maximal curves
James W. P. Hirschfeld and Ga´bor Korchma´ros
Dedicated to Adriano Barlotti on the occasion of his 80th birthday
Abstract.A lower bound for the size of a complete cap of the polar spaceHðn;q2Þassociated to the non-degenerate Hermitian varietyUnis given; this turns out to be sharp for evenqwhen n¼3. Also, a family of caps ofHðn;q2Þis constructed fromFq2-maximal curves. Such caps are complete forn¼3 andqeven, but not necessarily forqodd.
1 Introduction
Let Un be the non-degenerate Hermitian variety of the n-dimensional projective space PGðn;q2Þ coordinatised by the finite field Fq2 of square order q2. An ovoid of the polar space Hðn;q2Þ arising from the non-degenerate Hermitian variety Un
withnd3 is defined to be a point set in Un having exactly one common point with every generator of Un. For n even, Un has no ovoid; see [23]. For n odd, the exis- tence problem for ovoids ofUnhas been solved so far only in the smallest casen¼3;
see [25].
A natural generalization of an ovoid is a cap (also called apartial ovoid). A cap ofUnis a point set inUnwhich has at most one common point with every generator ofUn. Equivalently, a cap is a point set consisting of pairwise non-conjugate points ofUn. A cap is calledcompleteif it is not contained in a larger cap ofUn.
The size of a cap is at mostqnþ1 for oddnandqnfor evenn; equality holds if and only if the cap is an ovoid. The following upper bound for the sizekof a cap di¤erent from an ovoid is due to Moorhouse [19]:
kc pþn1 n 2
pþn2 n 2
" #h
þ1; q¼ph: ð1:1Þ
A lower bound forkis given in Section 2 by proving thatkdq2þ1.
In this paper a family of caps of Un that are not ovoids is constructed, and it is shown that they are complete provided that n¼3 and q is even. The construction relies on an interesting property ofFq2-maximal curves of PGðn;q2Þthat is stated in
§3: theFq2-rational points of anFq2-maximal curve naturally embedded in a Hermi- tian variety Un are pairwise non-conjugate under the associated unitary polarity.
Hence the setXðFq2Þof allFq2-rational points of anFq2-maximal curveX is a cap ofUn. The main result is thatXðFq2Þis a complete cap forn¼3 andqeven.
Forn¼3 andqodd, there existFq2-maximal curves such thatXðFq2Þis a cap of size12ðq3qÞcontained in an ovoid ofU3; see Example 4.8.
2 A lower bound for the size of a complete cap ofUn
A (non-degenerate) Hermitian variety Un is defined as the set of all self-conjugate points of a non-degenerate unitary polarity of a projective space PGðn;q2Þ. Hermi- tian varieties of PGðn;q2Þare projectively equivalent, as they can be reduced to the canonical form
X0qþ1þ þXnqþ1¼0
by a non-singular linear transformation of PGðn;q2Þ. AgeneratorofUnis defined to be a projective subspace of maximum dimension lying on Un, namely of dimension
1 2ðn1Þ
. General results on Hermitian varieties are due to Segre [21]; see also [15], [14], [16]. Here, some basic facts from [16, Section 23.2] are recalled. Let mn denote the number of points onUn.
Result 2.1.(1)mn¼ ðqnþ1þ ð1ÞnÞðqn ð1ÞnÞ=ðq21Þ.
(2)For any point PAUn,the number of lines through P and contained inUnis equal tomn2.
(3)The tangent hyperplane at PAUn meetsUnin q2mn2þ1points.
Now we give a lower bound for the size of complete caps which does not depend onn.
Theorem 2.2.The size k of a complete cap ofUnsatisfies kdq2þ1.
Proof. The assertion is true for ovoids. LetKbe a complete cap ofUn that is not an ovoid. Take a generatorH ofUn disjoint fromK. For any pointPAK, the tangent hyperplanePPtoUn atPdoes not containH. In fact, some point ofH is not con- jugate toP, as H is a projective subspace of maximum dimension contained inUn. This implies thatPPVH is a hyperplaneHðPÞofH. AsKis a complete cap ofUn, the projective subspaces HðPÞcoverH asPranges overK. SinceH is a projective space of dimensionr¼12ðn1Þ
, this yields
1þq2þ þq2rckð1þq2þ þq2ðr1ÞÞ:
Hence
kdq2þ1=ð1þq2þ þq2ðr1ÞÞ:
Sincekis an integer, this is only possible forkdq2þ1. r
The above lower bound is sharp forn¼3 and evenq; see Example 3.6 and Theo- rem 4.1 with g¼0. For the classification of transitive ovoids when n¼3 and q is even, see [5]. It is not known whether the lower bound is sharp forn>3 or forn¼3 and arbitrary oddq. To the best of our knowledge, the smallest complete cap ofUn
for anyqis that described in the following theorem.
Theorem 2.3.Leta be a plane ofPGðn;q2Þwhich meetsUnin a non-degenerate Her- mitian curveU2.ThenU2 is a complete cap ofUnof size q3þ1.
Proof. First, U2 is a cap of Un. Let AAUn be any point. The tangent hyperplane PAtoUatAeither containsaor meets it in a linel. It turns out in both cases that PAhas a common point withU2, whence the assertion follows. r
3 Hermitian varieties and maximal curves
In algebraic geometry in positive characteristic the Hermitian variety is defined to be the hypersurfaceUnof homogeneous equation
X0qþ1þ þXnqþ1¼0;
viewed as an algebraic variety in PGðn;FÞwhere Fis the algebraic closure ofFq2. Points ofUn are the points ofUnwith coordinates inFq2, usually calledFq2-rational points ofUn. For a pointA¼ ða0;a1;. . .;anÞofUn, the tangent hyperplane toUn at Ahas equation
a0qX0þa1qX1þ þanqXn¼0:
In this paper, the term algebraic curve defined over Fq2 stands for a projective, geometrically irreducible, non-singular algebraic curve X of PGðn;q2Þ viewed as a curve of PGðn;FÞ. Further, XðFq2iÞ denotes the set of points ofX with all coor- dinates in Fq2i, calledFq2i-rational points of X. For a point P¼ ðx0;. . .;xnÞof X, the Frobenius image of P is defined to be the point FðPÞ ¼ ðx0q2;. . .;xnq2Þ. Then P¼FðPÞif and only ifPAXðFq2Þ.
An algebraic curve X defined over Fq2 is called Fq2-maximal if the number Nq2
of its Fq2-rational points attains the Hasse–Weil upper bound, namely Nq2¼q2þ 1þ2gq, where gdenotes the genus of X. In recent years,Fq2-maximal curves have been the subject of numerous papers; a motivation for their study comes from coding theory based on algebraic curves having many points over a finite field. Here, only results on maximal curves which play a role in the present investigation are gathered.
Result 3.1 (Natural embedding theorem [17]). Up to Fq2-isomorphism, the Fq2- maximal curves ofPGðn;q2Þare the algebraic curves defined overFq2 of degree qþ1 and contained in the non-degenerate Hermitian varietyUn.
Remark 3.2.TheFq2-maximality ofXimplies thatðqþ1ÞP1qQþFðQÞfor every QAX, and the natural embedding arises from the smallest linear series S contain-
ing all such divisors. Apart from some exceptions, S is complete and hence S¼ jðqþ1ÞP0jfor anyP0AFq2. By the Riemann–Roch theorem, dimS¼qþ1gþi where i is the index of speciality. In many situations, for instance when qþ1>
2g2, we havei¼0, and hence dimS¼qþ1g. With our notation,n¼dimS.
This, together with some more results from [17], gives the following.
Result 3.3. Let X be an Fq2-maximal curve naturally embedded in Un. For a point PAX,letPPbe the tangent hyperplane toUnat P.ThenPPcoincides with the hyper- osculating hyperplane toXat P,and
PPVX¼ fPg for PAXðFq2Þ;
fP;FðPÞg for PAXnXðFq2Þ:
ð3:1Þ
More precisely,for the intersection divisor D cut out onXbyPP,
D¼ ðqþ1ÞP for PAXðFq2Þ;
qPþFðPÞ for PAXnXðFq2Þ:
ð3:2Þ
Theorem 3.4. LetX be anFq2-maximal curve naturally embedded inUn.For a point AAUnnX,letPAbe the tangent hyperplane toUnat A.If n¼3and q is even,thenPA
has a common point withXðFq2Þ.
Proof.Letlbe a line ofUn. Thenl, viewed as a line of PGðn;FÞ, is contained inUn. LetQAlVX; then it must be shown thatQAXðFq4Þ.
Assume, on the contrary, that QAXðFq2iÞ with id3. Then the three points Q;FðQÞ;FðFðQÞÞare distinct points ofX. Sincel is defined overFq2, solcontains not only Qbut alsoFðQÞ andFðFðQÞÞ. By (3.1), the hyper-osculating hyperplane PQ toX at QcontainsFðQÞ, and hencePQ contains the line l. But thenPQ must containFðFðQÞÞ, contradicting (3.1).
Assume now thatQAXðFq4Þ. The previous argument also shows that lcontains bothQandFðQÞbut no more points fromX. Also,lcannot contain more than one point fromXðFq2Þ, again by (3.1). Hence, iflVXis non-trivial, then eitherlVXis a singleFq2-rational point orlVXconsists of two distinct points, Frobenius images of each other, both inXðFq4ÞnXðFq2Þ.
LetQAUn be any point inPAVX. Then the linelthroughAandQis contained in Un. Now, assume thatn¼3; then such a line is contained in Un. By the above assertions, the points inPAVXareFq4-rational points ofX. For a pointQAX, let IðX;PA;QÞdenote the intersection multiplicity ofXandPA atQ. By Be´zout’s the- orem,P
QIðX;PA;QÞ ¼qþ1 whereQranges over all points ofX. Write X
Q
IðX;PA;QÞ ¼X
Q
0IðX;PA;QÞ þX
Q
00IðX;PA;QÞ;
where the summation P0
is over XðFq2Þ while P00
is over XðFq4ÞnXðFq2Þ. Since bothPAandXare defined overFq2,
IðX;PA;QÞ ¼IðX;PA;FðQÞÞ:
HenceP00
QIðX;PA;QÞ1qþ1 ðmod 2Þ. Forq even, this implies thatIðX;PA;QÞ>
0 for at least one pointQAXðFq2Þ, whence the assertion follows. r Remark 3.5. Theorem 3.4 might not extend to n>3. For a point AAUn, let QAUn be a point other than Ain the tangent hyperplanePA ofUn atA. Ifn¼3, then the linelthroughAandQisFq2-rational. But this assertion does not hold true forn>3.
In fact, letUn be given in its canonical form
X0qXnþX0XnqþX1qþ1þ þXn1qþ1¼0:
It may be assumed that A¼ ð0;. . .;0;1Þ. Then PA has equation X0¼0 and Q¼ ð0;a1;. . .;an1;1Þwith a1qþ1þ þan1qþ1¼0. The linel isFq2-rational if and only ifFðQÞalso lies onl. This happens whenaiq2¼lai,i¼1;. . .;n1, for a suitable elementlAF, or, equivalently, when aiq21¼ajq21 for alli;j with 1ci;jcn1 andai;aj00. Now,a1qþ1¼ a2qþ1 impliesða1qþ1Þq1¼ ða2qþ1Þq1, whence the asser- tion follows for n¼3. Unfortunately, as soon as n>3, a1qþ1þ þan1qþ1¼0 does not imply aiq21¼ajq21 for any i;j with 1ci;jcn1 and ai;aj00. Thus the assertion is not valid forn>3.
The following example illustrates property (3.1).
Example 3.6.Still withqeven, write the equation ofU3in the form X0qX3þX0X3q¼X1qþ1þX2qþ1:
The rational algebraic curveXof degreeqþ1, consisting of all points AðtÞ ¼ fð1;t;tq;tqþ1Þ jtAFg
together with the pointAðyÞ ¼ ð0;0;0;1Þ, lies onU3. The morphism ð1;tÞ ! ð1;t;tq;tqþ1Þ
is a natural embedding. We note that the tangent hyperplanePAðtÞ toU3atAðtÞhas equation
tqðqþ1ÞX0þX3þtqX1þtq2X2¼0:
To show that (3.1) holds forAðtÞ, it is necessary to check that the equation tqðqþ1Þþuqþ1þtquþtq2uq¼0
has only two solutions in u, namely u¼t and u¼tq2. Replacing u by vþt, the equation becomes vqþ1þvqtþtq2vq¼0. For v00, that is, for u0t, this implies v¼tq2þt, proving the assertion. ForAðyÞ, the tangent hyperplanePAðyÞhas equa- tion X0¼0. Hence it does not meetX outside AðyÞ, showing that (3.1) also holds forAðyÞ.
4 Caps of the Hermitian variety arising from maximal curves From the results stated in Section 3 we deduce the following theorem.
Theorem 4.1.LetXbe anFq2-maximal curve naturally embedded inUn.Then (i) XðFq2Þis a cap ofUnof size q2þ1þ2gq;
(ii) when q is even and n¼3,such a cap is complete.
Proof.LetPAXðFq2Þ. By (3.1), no further point fromXis inPP. Hence no point in XðFq2Þis conjugate toP. This shows thatXðFq2Þis a cap ofUnwhose size is equal to q2þ1þ2gqby theFq2-maximality ofX. Completeness for evenqandn¼3 follows
from Theorem 3.4. r
In applying Theorem 4.1 it is essential to have information on the spectrum of the generag ofFq2-maximal curves. However, it would be inappropriate in the present paper to discuss the spectrum in all details; so we shall content ourselves with a summary of the relevant results in characteristic 2. For this reason, q will denote a power of 2 in the rest of the paper, apart from Example 4.8.
Result 4.2.(1)The lower limit of the spectrum of genera is0,which is only attained by rational algebraic curves.
(2)The upper limit of the spectrum is12ðq2qÞ,which is only attained by the Her- mitian curve overFq2;see[22, Proposition V.3.3].
Result 4.3 ([1], [10], [17]). (1)The second largest value in the spectrum of genera is
1
4ðq22qÞ,which is only attained by Example4.5.
(2)In the interval18ðq24qþ3Þ;14ðq2qÞ
,there are12known examples.
Result 4.4([18]).The third largest value in the spectrum is16ðq2qþ4Þ
.Examples 4.6and4.7are the only known examples with this genus.
Example 4.5([9]). The absolutely irreducible plane curveCwith equation yþy2þ þyq=2þxqþ1¼0
has genus14qðq2Þ. A non-singular modelXofCis theFq2-maximal curve defined by the morphismp:C!PGð3;q2Þwith coordinate functions
f0¼1; f1 ¼x; f2¼y; f3¼x2: The curveXlies on the Hermitian varietyU3with equation
X2qX0þX2X0qþX1qþ1þX3qþ1¼0:
Also, X lies on the quadric cone with equationX3X0¼X12. The size of the corre- sponding complete capXðFq2ÞofU3is12ðq3þ2Þ.
Example 4.6 ([7, Theorem 2.1. (IV)(2)]). Let q12 ðmod 3Þ. The absolutely ir- reducible plane curve C with equation xðqþ1Þ=3þx2ðqþ1Þ=3þyqþ1¼0 has genus g¼16ðq2qþ4Þ. A non-singular model X of C is the Fq2-maximal curve defined by the morphismp:C!PGð3;q2Þwith coordinate functions
f0¼x; f1¼x2; f2¼y3; f3¼x y:
The curveXlies on the Hermitian varietyU3given by the usual canonical equation X0qþ1þX1qþ1þX2qþ1þX3qþ1 ¼0:
Also,Xlies on the cubic surface with equation X33þw3X0X1X2¼0
with wqþ1¼ 3. The size of the corresponding complete cap XðFq2Þ of U3 is
1
3ðq3þ2q2þ4qþ3Þ.
Example 4.7([6, §6]). A similar but non-isomorphic example is given in [6]. Again, assume thatq12ðmod 3Þ. The absolutely irreducible plane curve Cwith equation
yxðq2Þ=3þyqþxð2q1Þ=3¼0
has genus 16ðq2q2Þ. A non-singular model X of C is the Fq2-maximal curve defined by the morphismp:C!PGð3;q2Þwith coordinate functions
f0¼x; f1¼x2; f2¼ y3; f3¼ 3x y:
The curveXlies on the Hermitian varietySqþ1 with equation X0qX1þX1qX2þX2qX03X3qþ1¼0:
Also,Xis contained in the cubic surface with equation X33þ27X0X1X2¼0:
It is worth noting that Sqþ1 is projectively equivalent to U3 in PGð3;q6Þ but not in PGð3;q2Þ. Nevertheless, the projective transformation takingSqþ1 toU3 maps X to an Fq2-maximal curve lying on U3. The size of the corresponding complete cap XðFq2ÞofU3 is13ðq3þ2q22qþ3Þ.
We end the paper with an example for q odd which shows that assertion (ii) in Theorem 4.1 does not hold forqodd.
Example 4.8.Let q be odd and letCðFq2Þbe the absolutely irreducible plane curve with equation
yqþyþxðqþ1Þ=2¼0;
it has genus 14ðq1Þ2. A non-singular model X of C is the Fq2-maximal curve defined by the morphismp:C!PGð3;q2Þwith coordinate functions
f0¼1; f1¼x; f2¼ y; f3 ¼y2: The curveXlies on the Hermitian surfaceU3with equation
X3qX0þX3X0qþ2X2qþ1X1qþ1¼0:
Also, Clies on the quadric cone Q with equation X22X0X3 ¼0. The size of the corresponding cap K of U3 is q2þ1þ12qðq1Þ2 ¼12ðq3þqþ2Þ. The cap K is incomplete, since it is contained in an ovoid ofU3; see [13].
Acknowledgements.The second author’s research was carried out within the project
‘‘Strutture geometriche, combinatoria e applicazioni’’ PRIN 2001–02, MIUR.
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Received 10 December, 2002; revised 18 March, 2003
J. W. P. Hirschfeld, School of Mathematical Sciences, University of Sussex, Brighton BN1 9QH, United Kingdom
Email: [email protected]
G. Korchma´ros, Dipartimento di Matematica, Universita` della Basilicata, 85100 Potenza, Italy
Email: [email protected]
