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On the intersection of Hermitian surfaces

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Šb† ¤K“…†E‚`‚dƒ„# z•‹†d„ Y‰Ÿ “…†f‰/–‚Xƒ„

GF(√q)

Œ;ŽŠY}„e•‹”`†E¢Ÿ†E‚d„˜¬‚dƒ„e¢’‹„e}„”S•'‚X„‹Ÿ}…•‹˜œ„Y¢

Γ(

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1

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2

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{H

1

+ λ

H

2

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∈ GF(

q)

}.

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Γ

•‹}…žžŸ„†d”dŠY z„=‚Xƒ„•oQ‚X“…•/ˆ"‰‹ŠY}.‚KŒ;ŽŠY}„—‰‹}Ÿ‘…’o“”S•'‚dŠY‰‹}z†®•/”XŠY†dŠŽ}…’#ŠŽ}œ‚Xƒ„

3

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Γ

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H

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rad

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H

n »¼„ ”X„ee•/Y"‚dƒ…•/‚2‚Xƒ„œµd¶'½Ÿ¾‰/–j•¡›f„e”d˜œŠŽ‚dŠb•/}‡4•‹”dŠY„ ‚K¢œŠY}

PG(n, q)

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r = n + 1

− dim rad H

n ¿s„‚l}‰'š

H

1

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H

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PG(n, q)

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r

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i

ŠY}§‚Xƒ„

GF(√q)

Œ.ˆ"„}z ŠY

Γ

‚Xƒ„¢ª’‹„e}„”S•'‚X„ n »¼„†Eƒz•/Y¦•/Y-‚dƒ…„¡YŠb†K‚

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2

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•‹†=ŠŽ‚fƒ…•‹†= "„„e}[ž‰‹}„ŠY}¼~Ç/€ n ÈÉ'ʅË2ÊzÌÍ+ÎÍNÊ"ÏÑÐzÒ[Ó"ԅÂ#µd¶/½;¾ÁÂSÃą ½u¹XÂ

(r

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n

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ÕEÖ ¶ ׏½"¹¸]º

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ÕKÖ`Ø ÂµQÙ¸]ÚM¸N¶'½Û'¶'µQ¸NÂÚM¸NÂQÁ#¸]½

PG(n, q)

Á¶'ÚM¸bÁ+Ü2 ÁÚ+ԏÂ?¸]½"ÂSÃą¶'ºa¸]ÚMÝ

n

X

i=1

(n

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i

≤ n + 1.

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H

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ŠY†e

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(2)

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H

2

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PG(n, q)

/ ¶/½"·¼ºŽÂÚ

(r

1

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n

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0 ÂÚ+ԏÂ?µd¶'½Ÿ¾¡ÁÂSà ąÂ½"¹S ÕKÖ Ú+ԏÂ`׏½"¹¸]º

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1

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H

2

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7 Ó"ԅ ½ /

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1

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2

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n

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1

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{(1 − q

n+1

) +

n

X

i=1

r

i

[(q

qµ(i

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− (q − 1)µ(n, q)]} + (1 +

1

q

)µ(n, q).

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PG(3, q)

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r

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r

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r

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η

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(Γ, q)

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1

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q√q + q + 1

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q√q + 1

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1

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Γ

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q + 1

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C

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3

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q + 1

‰‹”

q + 1

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n =

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•‹}…ž žŸ„}‰‹‚d„ ;¢

v

2

‚Xƒ„#};“˜? "„”`‰‹–-‰‹˜œˆz‰o}„}.‚S†=‰/–

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š`ƒŠbSƒ˜œ„e„ ‚

E

ŠŽ}[•žŸ„’o„}„e”X•/‚d„›f„e”d˜œŠŽ‚dŠb•/}[ “”X‡‹„ n

(3)

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q

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n ´=ƒ„e}s

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q + q

q) +

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q + 1

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q + 1;

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q + 1

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2

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q

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ƒ„}…„‹

v

2

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n



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P

0

•/”X„#•/};¢‚Kšk‰žŸ„’o„}„e”X•/‚d„†E“”d–N•‹„e†kŠY}

Γ

Ÿ‚dƒ…„}

R = rad P ∩

rad

P

0

=

†dŠY}… „o‹‰/‚Xƒ„”Xš`Šb†E„o‹•/};¢ˆ"‰‹ŠY}.‚

V

∈ R

šk‰‹“bž¡ z„†EŠY}’‹“…Y•‹”-–+‰‹”¦•/YŸ‚Xƒ„—†d“”E–N•o „†-‰/–

Γ

n Þ “ˆˆ"‰o†d„ }‰'š ‚dƒz•'‚—‚Xƒ„”X„•‹”d„‚Kšk‰žŸŠb†E‚dŠY}…Q‚#›f„e”d˜œŠŽ‚dŠb•/}ņd“”E–N•o „†

P

•/}…ž

P

0

 "‰/‚dƒ§‰/–¦”S•/}À

2

ŠY}

Γ

~ ‚dƒ„ˆ”X„‡;ŠŽ‰o“…† ‰‹ z†E„e”d‡'•'‚XŠŽ‰o}#ˆ”X‰'‡‹„e†s‚dƒ…•/‚

rad

P

•‹}…ž

rad

P

0

ƒ…•4‡‹„¦‚d‰  z„k˜?“Ÿ‚X“…•/YY¢†EÀo„š nK “”E‚Xƒ„”X˜œ‰‹”X„‹4 "‰/‚dƒ

rad

P

•/}…ž

rad

P

0

˜œ„„‚—•/};¢}…‰‹};ŒŸžŸ„’o„}„e”X•/‚d„†E“”d–N•‹„#ŠŽ}

Γ

ŠY}

(√q + 1)

ˆ"‰‹ŠY}o‚S† n ´=ƒ;“…†…š2„#‰o Ÿ‚X•‹ŠŽ}‚dƒ…„#–+‰‹YŽ‰'š`ŠY}’ ˆ”X‰‹ˆ"‰o†dŠ‚XŠŽ‰o} n ÈÉ'ʅË2ÊzÌÍ+ÎÍNÊ"ρ€ÒÓuԏÂ#¸]½zÚiµSÁÂS¹ ÚM¸ Õ ½ ÕEÖ Ú21 Õ ½ Õ ½§·‹Â65. ½u µX¶'Ú6Â Ø ÂµQÙ¸]ÚM¸N¶'½ÅÁSğµ Ö ¶o¹SÂQÁ

H

1

/

H

2

ÁM׏¶‚1®½…¸]½Y5¶ ׅ ½u¹ ¸]º91®¸]Ú+Ô

r

2

= 2

¸bÁ¡Ú+ԏ¡ğ½z¸ Õ ½ ÕKÖ ¶'º]ºK5.½"µd¶'Ú Õ µSÁ ÕKÖ

H

1

1jÔ;¸N¹SÔׅ¶4ÁSÁÚ+Ô;µ Õ Ä]54ÔªÚ1 Õ Ád¾oÂl1

(√q + 1)

3 ÁÂS¹X¶/½…ÚCÁ 7 £ n£ n ȃN.ÏO‹ÍPNÌaR„S)ÊzÌNbV_NYX=N;Ï_N.É*TŸÎEN§ÌE[jÉ*\2T4O]N;Ì.ST]c=NÅÉ*TÏ_^

2

TÏV

3

Ò © ŠŽ‡o„}[•›f„e”d˜œŠŽ‚dŠb•/} ‰o}„

C

‰/– ‡‹„e”E‚X„l…

V

•‹}…ž•}…‰‹};ŒŸžŸ„’o„}„e”X•/‚d„›f„”X˜œŠ‚XŠY•‹}[‡4•‹”dŠY„ ‚K¢

H

š2„#žŸ„e}‰/‚X„# ;¢

Γ

0

(

C, H)

‚Xƒ„

GF(√q)

Œ;ŽŠY}„•/” †d¢;†E‚d„e˜ ‰/–-›f„”X˜œŠ‚XŠY•‹}“”d‡o„e†2’‹„e}„”S•'‚X„ež ;¢

C

0

=

C ∩ π

•‹}…ž

H

0

=

H ∩ π

š`ƒ„e”d„

π

Šb†k‚dƒ„ˆ"‰‹b•/”kˆY•‹}„ ‰/–

V

š`Š‚Xƒ[”d„†Eˆ"„e ‚`‚d‰

H

n †‡NYˆˆ‰T‹ŠsÒa-j Ú

C

1

¶/½"·

C

2

0 ÂÚ1 Õ ·'¸bÁQÚM¸]½"¹Ú Ø Â µQÙ¡¸]ÚM¸N¶'½Ñ¹ Õ ½" Á ÕKÖ Û'µQÚC¸N¹S ÁµdÂQÁMׅÂS¹ ÚM¸]Û4ºaÝ

V

1

¶'½u·

V

2

7 Œ ÁSÁQğÙ—Ú+ԏ¶/Ú®Ú]ԅÂl׏½"¹¸]º

Γ(

C

1

,

C

2

)

¹ Õ ½zÚi¶/¸]½Á ¶/Ú®ºŽÂS¶4ÁQÚ¦¶?½ Õ ½Z3/ÁQ¸]½"5‹ÄŸºŽ¶'µÁQğµ Ö ¶‹¹SÂ#¶'½u·Ú]ԅ¶'Ú

V

1

6∈ E

7 Óuԏ½ /

V

2

0 º Õ ½Y5'Á#Ú Õ Ú+ԏÂk× Õ ºY¶'µ=×uºŽ¶'½u ÕEÖ

V

1

1®¸]Ú+Ô[µd ÁM׏ÂS¹Ú2Ú Õ ¶'½zݽ Õ ½43.·oÂ65o½"µd¶'Ú6Â Ø ÂµQÙ¸]ÚM¸N¶'½§ÁQğµ Ö ¶‹¹S¸]½

Γ

7 kkµ ÕÕKÖl7  Š…§•}‰o};Œ;žŸ„e’‹„e}„”S•'‚X„›f„”X˜œŠ‚XŠY•‹}ª†d“”d–N•‹ „

H ∈ Γ

•/}…ž§Ž„‚

π

 "„‚dƒ„¡ˆz‰oY•‹”ˆ…Y•‹}„?‰‹–

V

1

š`ŠŽ‚dƒ ”X„e†dˆz„Q‚‚d‰

H

n Þ ŠŽ}z „

V

1

6∈ H

^‚Xƒ„œˆY•‹}„

π

“Ÿ‚X†#•}‰‹}ŸŒ;†dŠŽ}’o“b•/”›f„”X˜œŠ‚XŠY•‹}ª“”d‡o„‰o}

H

n Þ “ˆ…ˆz‰.†E„

V

2

6∈ π

~ ‚dƒ…„}¡‚dƒ„=YŠŽ}…„

V

1

V

2

šk‰‹“bž˜¡„e„ ‚

H

ŠŽ}

q +1

ˆ"‰‹ŠY}.‚X† n } }‚dƒ…„`‰/‚dƒ…„”-ƒ…•/}zž^

V

1

V

2

˜œ„„‚X†

C

1

•/}…ž

C

2

ŠŽ}[„eŠ‚Xƒ„”

1

‰‹”

q + 1

ˆ"‰‹ŠY}.‚X†Ž •œ ‰o}.‚d”S•‹žŸŠbQ‚XŠŽ‰o} n |i‚`–+‰oŽY‰'šf†k‚dƒz•'‚

V

2

∈ π

•‹}…ž

|V

1

V

2

∩ E| ≤ 1

n



†‡NYˆˆ‰ThÒӅ¶¾‹Â

C

Ú Õ 0 Â#¶ Ø Â µQÙ¡¸]ÚC¸N¶/½¹ Õ ½" ÕKÖ Û4µQÚ6Âi

V

¶'½u·

H

Ú Õ 0 Â#¶½ Õ ½43;·‹Â65o½"µd¶/ÚiÂ Ø Â µQÙ¡¸]ÚC¸N¶/½ ÁQğµ Ö ¶‹¹SÂllºŽÂÚ=¶/ºHÁ Õ

π

0 ÂÚ+ԏÂ`× Õ ºŽ¶/µ=×"ºŽ¶/½" ÕKÖ

V

1®¸]Ú]Ôµd ÁCׅÂS¹ Ú2Ú Õ

H

¶'½u·ºŽÂÚ

Γ = Γ(

C, H)

7 Óuԏ½ / { m$| ¸ Ö

V

6∈ H

/

η

3

(Γ, q) = q

2

+ q

q +

q + 1

− η

2

0

, q)

q;

{  | ¸ Ö

V

∈ H

/

η

3

(Γ, q) = q

2

− q + |π ∩ E|.

kkµ ÕÕKÖl7 ¿s„‚

H

0

=

H ∩ π

•‹}…ž

C

0

=

C ∩ π

n ´j•‹À‹„

h = η

2

0

, q)

n }  …†E„e”d‡o„‚Xƒ…•'‚•/};¢YŠŽ}…„‚Xƒ”X‰‹“’oƒ

V

‚X•‹}’‹„e}.‚k‚X‰

H

Šb†=‰/–‚Xƒ„ –+‰‹”X˜

P V

š`Š‚Xƒ

P

∈ H

0

n |i–

V

6∈ H

Ÿ„‡o„”X¢ŽŠY}„ ‚dƒ”X‰‹“…’‹ƒ

V

˜œ„„‚X†

H

ŠŽ}„ŠŽ‚dƒ„e”

1

‰‹”

q + 1

ˆ"‰‹ŠY}.‚X† ~ ‰o}§‚Xƒ„‰/‚Xƒ„”#ƒ…•‹}…ž^j„l…•‹ ‚dY¢

h

’‹„e}„”S•'‚X‰‹”S†—‰‹–

C

•/”X„¡‚X•‹}’‹„e}o‚‚X‰

H

š`ƒ„e}… „œŠŽ‚ –+‰‹YY‰'šf†

η

3

(Γ, q) = h + (q

q + 1

− h)(

q + 1).

8 †X†d“˜œ„

V

∈ H

n ›f„}z „‹

π

Šb†®‚dƒ…„f‚X•‹}’‹„e}.‚lˆb•/}„`‚X‰

H

•'‚

V

•/}…ž

H

0

‰‹}…†dŠb†K‚S†¦‰/–

q + 1

ŽŠY}„†¦‚Xƒ”d‰o“’‹ƒ

V

n ›f‰'šk„‡o„”

C

0

‰‹}…†dŠb†K‚S†=‰/–-„ŠŽ‚dƒ„e”

1

ŽŠY}„‰o”f•¡ž„’‹„e}„”S•'‚X„#›f„”X˜œŠ‚XŠY•‹} “”X‡‹„ n |6}[‚dƒ„–+‰o”d˜œ„”fe•‹†d„š2„

(4)

š2‰o“YžÅƒ…•4‡‹„

η

3

(Γ, q) = q

2

+ q + 1

uš`ƒŠbSƒÅŠb† }‰/‚ ˆz‰.†d†dŠŽ …Ž„ n ›—„}…„‹

C

0

ŠY†—‚dƒ…„“}…ŠŽ‰o}§‰/–

q + 1

YŠY}„e† ‚dƒ”X‰‹“’oƒ

V

•‹}…ž

η

3

(Γ, q) =

q(q

q

q) +

|π ∩ E|.

}  …†E„e”d‡o„—‚Xƒ…•'‚fŠY}‚dƒŠb†f•o†E„o•/Yu‚dƒ„“”X‡‹„e†=ŠY}‚dƒ„#YŠŽ}…„e•/”f†d¢Ÿ†K‚X„˜

Γ

•‹”d„#žŸ„e’‹„e}„”S•'‚X„ n



’—†dŠŽ}’Æ‚dƒ„¼•‹”XžŸŠY}…•‹ŽŠŽ‚K¢Æ–+‰o”d˜“b• ‰/– B ”d‰oˆz‰.†EŠŽ‚dŠY‰‹} Ÿf‚d‰o’‹„‚dƒ„e”š`ŠŽ‚dƒ ¿„˜œ˜•‰“…—ŠŽ‚Šb†ˆ"‰o†X†dŠŽ Y„Å‚X‰

”d„ ‰‹}z†K‚X”d“… ‚2‚Xƒ„#”S•/}À†d„e¥.“„e}… „‰‹–

Γ(

C, H)

–+”X‰‹˜ ‚Xƒ„#”X•‹}À†d„e¥.“„e}… „‰/–

Γ

0

n †‡NYˆˆT•”sÒ Œ ÁXÁQğÙÂ

Γ

Ú Õ ¹ Õ ½zÚi¶/¸]½ª¶'Ú2ºŽÂS¶4ÁQÚ Õ ½"¹ Õ ½"Â

C

ÕKÖ Û'µQÚiÂji

V

6∈ E

/ ¶'½u·œºŽÂÚlÚ+ԏÂ?µd¶'½Ÿ¾¡ÁÂSà ąÂ½"¹S ÕKÖ

Γ

0

0 Â

(r

0

1

, r

0

2

)

7 Óuԅ ½ / Ú]ԅÂ#µd¶/½;¾ÁÂXÃąÂ½"¹S ÕEÖ

Γ

¸bÁ

(0, r

0

1

, r

0

2

+ 1)

7 |6}–N•‹ ‚e;Š‚=Šb†2ˆ"‰o†X†dŠŽ Y„f‚X‰žŸ„†d”dŠY "„—ŠY}•/}•‹“”S•'‚d„—šk•4¢¡‚dƒ…„ •oQ‚d“z•/u ‰‹}‘…’‹“”S•'‚XŠŽ‰o}

E ∩ π

ŠŽ}‚d„”X˜†l‰‹– ‚dƒ„†d„‡o„}[ b•‹†X†E„†2‰‹–2~H4€n »ª„†dƒ…•‹ŽsžŸ„e}‰/‚X„†E“zSƒY•o†d†d„e†=•‹†=ŠY}«~£'€n ÈÉ4ÊzËlÊ"ÌeÍNÎeÍCÊzϗ–sÒ-jÂÚ

Γ

0 Â`¶

GF(√q)

˜ ׅ ½u¹ ¸]º ÕKÖ-Ø ÂµQÙ¸]ÚM¸N¶'½¡ÁQğµ Ö ¶‹¹S Á‡1®¸]Ú+ÔµX¶'½;¾—ÁÂSÃą ½u¹XÂ

(0, r

0

1

, r

2

0

+1)

7 Ó"ԅ ½ / Ú+ԏ 0 ¶'Á¹ Õ ½Ü_5‹ÄŸµd¶'ÚM¸ Õ ½

E

0

=

E ∩ π

¸bÁ#ğ½…¸NÃąÂºaÝ·‹ÂÚiµQÙ¡¸]½"ÂS· 7 kkµ ÕÕKÖl7 ´=ƒ„•/”SžŸŠY}…•/YŠŽ‚K¢‰‹–

E

0

Šb†#žŸ„‚d„”X˜œŠŽ}…„ežª .¢Å¿„˜œ˜•—™ n }  …†d„”X‡‹„?‚dƒ…•/‚

5

‰‹–¦‚dƒ…„

7

 b•‹†X†d„e†‰/–~H€ •/”X„“}Šb¥.“„Y¢ž„ ‚d„e”d˜œŠY}„ežÅ .¢‚dƒ…„ŠY””S•/}À[†d„e¥.“„}z „ n ›—‰'š2„e‡‹„e”e "‰/‚Xƒ§ b•‹†X†f|d|E|f•‹}…ž|›š ‰‹”X”d„†Eˆ"‰‹}zž‚X‰ ‚dƒ„?†X•/˜œ„#”S•/}À†E„¥.“„}…„

(r

0

1

, r

0

2

) = (0, 1)

nœ ¢¿„˜œ˜•W™Ÿ

Γ

ƒ…•o†f}„ „e†X†X•/”XŠŽY¢”S•/}À†d„e¥.“„e}… „

(0, 0, 2)

•/}…ž

|E| = q

2

+ 1

nGž „e}‰/‚X„‚dƒ…„}Å ;¢

C

1

•/}zž

C

2

‚dƒ„?‚Kš2‰žŸŠY†E‚dŠY}… ‚ ›f„e”d˜œŠŽ‚dŠb•/}Ő‰‹}„†f‰/–

Γ

•/}…ž•o†d†d“˜œ„ ‚dƒ„e¢ƒ…•4‡o„”X„e†dˆz„Q‚XŠŽ‡o„Y¢‡‹„e”E‚XŠY„e†

V

1

•‹}…ž

V

2

n ´=ƒ„”X„#•/”X„ ‚Kš2‰œˆ"‰o†X†EŠY ŠYŽŠŽ‚dŠY„e†k–+‰o”

E

0

=

E ∩ π

D { m|

E

0

 "„Y‰‹}…’o†2‚d‰ b•‹†X†2|E|E|Q‚dƒ…•/‚=Šb†

E

0

‰‹}…†dŠY†E‚X†k‰‹–

q

− 1

†E“… ŽŠY}„†•‹ŽsžŸŠb†N¤K‰‹ŠY}o‚•/}…ž

2

˜œ‰o”d„ˆz‰oŠŽ}.‚X† ~ {  |

E

0

 z„eŽ‰o}’o†k‚X‰œY•o†d†=|›šŸ‚dƒz•'‚`ŠY†

E

0

 ‰o}…†dŠY†E‚X†=‰‹–

q

†d“ YŠY}„e†eš`ƒŠbSƒ˜œ„e„ ‚—•/Y^ŠY}[•¡ˆ"‰‹ŠY}.‚

P

n ´=ƒ„–+‰‹”X˜œ„”`•o†E„‰Ÿ“”X†kŠŽ–

V

2

6∈ E

n |i–

E

0

 "„Y‰‹}’.†k‚d‰ b•‹†X†=|›š?‚dƒ„e}

V

2

= P

∈ E

n



ÈÉ4ÊzËlÊ"ÌeÍNÎeÍCÊzÏ ŸÒ¡-j Ú

Γ

0 ¶§½ Õ ½43;·‹Â65o½"µd¶/Úiºa¸]½"ÂS¶/µÁQÝ4ÁSÚ6ÂÙ ÕEÖ?Ø ÂµQÙ¸]ÚM¸N¶'½ÆÁQğµ Ö ¶‹¹SÂQÁ„1®¸]Ú+Ô

r

2

(Γ) =

r

3

(Γ) = 1

7 Óuԏ½ /

E

¸bÁ` ¸]Ú+ԏµ2Ú+ԏÂ`ğ½…¸ Õ ½ ÕEÖ

q+1

½ Õ ½Z3.·oÂ25.½"µd¶'Ú6Â Ø ÂµQÙ¡¸]ÚC¸N¶/½¹ ğµQÛ'ÂQÁ`¶'º]ºs1®¸]Ú+Ԝ¶l× Õ ¸]½…Ú ¸]½Æ¹ Õ ÙÙ Õ ½ / Õ µ¡Ú]ԅÂğ½z¸ Õ ½ ÕKÖ

q

½ Õ ½Z3.·oÂ25.½"µd¶'Ú6Â Ø Â µQÙ¡¸]ÚC¸N¶/½Æ¹ ğµQÛ' Áœ¶'½u·Å¶·‹Â65. ½u µX¶'Ú6Â Ø ÂµQÙ¡¸]ÚC¸N¶/½ ¹ ğµQÛ' / ¶/º]ºsÁdԏ¶/µQ¸]½"5[¶

(√q + 1)

3‹ÁÂX¹S¶/½…Ú 7 kkµ ÕÕKÖl7 ¿s„ ‚

P

•/}zž

C

 z„ ”d„†Eˆ"„e ‚dŠY‡‹„eŽ¢¡‚dƒ…„‰o}Ž¢†d“”d–N•‹ „‰/–”S•/}…À

2

•/}…ž‚dƒ„ ‰‹}Y¢›f„e”d˜œŠŽ‚dŠb•/}‰‹}„ŠŽ}

Γ

n ¿s„ ‚—•‹Y†d‰

L = rad

P

•/}zž

V

 "„‚dƒ„#‡o„”d‚d„l…‰/–

C

n }  …†d„”X‡‹„‚dƒz•'‚

l =

|L ∩ E| ∈ {1,

q + 1

}

n { m|

l = 1

n ¿s„ ‚

M

 "„#‚Xƒ„?ˆ"‰‹ŠY}.‚—‰/–®ŠY}o‚X„”S†E„Q‚XŠŽ‰o}‰‹–

C

•/}zž

L

~  Y„e•/”XY¢

M

6= V

n |i–

V

∈ E

…‚Xƒ„}‚Xƒ„”X„ Šb†¡•§ ‰‹˜œˆ"‰‹}„e}.‚

π

‰‹–

P

†E“…Sƒ ‚dƒ…•/‚

V

∈ π

n ›—‰'š2„e‡‹„”)ŠŽ}¨‚XƒŠb†¡e•‹†d„

C ∩ π = P M

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