CaseM= 0
Proposition
M = 0 ⇐⇒ E splits into two distinct absolutely irreducible conics defined over a quadratic extension GF(q2) of GF(q) having no point in PG(2,q) and conjugate to each other over GF(q).
For p >2, this is the case⇐⇒
For p = 2, similar results hold.
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
CaseM= 0
Proposition
M = 0 ⇐⇒ E splits into two distinct absolutely irreducible conics defined over a quadratic extension GF(q2) of GF(q) having no point in PG(2,q) and conjugate to each other over GF(q).
For p >2, this is the case⇐⇒
For p = 2, similar results hold.
Proof Classification Ω ∩ U
CaseM= 0
Proposition
M = 0 ⇐⇒ E splits into two distinct absolutely irreducible conics defined over a quadratic extension GF(q2) of GF(q) having no point in PG(2,q) and conjugate to each other over GF(q).
For p >2, this is the case⇐⇒
For p = 2, similar results hold.
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
CaseM= 0
Proposition
M = 0 ⇐⇒ E splits into two distinct absolutely irreducible conics defined over a quadratic extension GF(q2) of GF(q) having no point in PG(2,q) and conjugate to each other over GF(q).
For p >2, this is the case⇐⇒
a11=γ12−sγ22, a12=α1γ1−sα2γ2, a22= 2γ1+12(α21−sα22−1s(β12−sβ22)), a23=α1,a33= 1, a13= 14(α21−sα22+1s(β21−sβ22)), b12= 1s(β1γ1−sβ2γ2), b13= 2s1(α1β1−sα2β2), b23= 1sβ1.
Example: α1 =α2=β1 =β2 = 0, γ1=γ2 = 1,
Proof Classification Ω ∩ U
CaseM>0
P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3. The goal is to computeM. For this purpose, the following cases are to be considered separately.
E is a line (this case cannot actually occur),
E is a conic, either absolutely irreducible or reducible; E is a cubic, either absolutely irreducible or reducible;
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
CaseM>0
P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.
The goal is to computeM. For this purpose, the following cases are to be considered separately.
E is a line (this case cannot actually occur),
E is a conic, either absolutely irreducible or reducible; E is a cubic, either absolutely irreducible or reducible;
Proof Classification Ω ∩ U
CaseM>0
P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.
The goal is to computeM. For this purpose, the following cases are to be considered separately.
E is a line (this case cannot actually occur),
E is a conic, either absolutely irreducible or reducible; E is a cubic, either absolutely irreducible or reducible;
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
CaseM>0
P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.
The goal is to computeM. For this purpose, the following cases are to be considered separately.
E is a line (this case cannot actually occur),
E is a conic, either absolutely irreducible or reducible; E is a cubic, either absolutely irreducible or reducible;
Proof Classification Ω ∩ U
CaseM>0
P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.
The goal is to computeM. For this purpose, the following cases are to be considered separately.
E is a line (this case cannot actually occur),
E is a conic, either absolutely irreducible or reducible;
E is a cubic, either absolutely irreducible or reducible;
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
CaseM>0
P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.
The goal is to computeM. For this purpose, the following cases are to be considered separately.
E is a line (this case cannot actually occur),
E is a conic, either absolutely irreducible or reducible;
E is a cubic, either absolutely irreducible or reducible;
Proof Classification Ω ∩ U
E is an absolutely reducible conic
Proposition
IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and2,2q for p= 2. More precisely, one of the following cases occurs:
(i) E =`0∪`1 with `0, `1, defined over GF(q),|`0∩`1|= 1;
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely reducible conic
Proposition
IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and 2,2q for p= 2. More precisely, one of the following cases occurs:
(i) E =`0∪`1 with `0, `1, defined over GF(q),|`0∩`1|= 1; Ω∩ U = Ω0∪Ω1,|Ω0∩Ω1|= 2;
(ii) E =`0∪`1 with `0, `1 defined over GF(q),`0 k`1; Ω∩ U = Ω0∪Ω1,|Ω0∩Ω1|= 1 p>2;
(iii) E =`0∪`1 with `0, `1 defined over GF(q),`0 =`1; Ω∩ U is a classical Baer suboval ofΩ, p>2;
(iv) E =`0∪`1 with `0, `1 defined over GF(q2),|`0∩`1|= 1; Ω∩ U ={P∞,A};
(v) E =`0∪`1 with `0, `1 defined over GF(q2),`0k`1; Ω∩ U ={P∞}; p >2.
Proof Classification Ω ∩ U
E is an absolutely reducible conic
Proposition
IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and 2,2q for p= 2. More precisely, one of the following cases occurs:
(i) E =`0∪`1 with `0, `1, defined over GF(q),|`0∩`1|= 1;
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely reducible conic
Proposition
IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and 2,2q for p= 2. More precisely, one of the following cases occurs:
(i) E =`0∪`1 with `0, `1, defined over GF(q),|`0∩`1|= 1;
Ω∩ U = Ω0∪Ω1,|Ω0∩Ω1|= 2;
(ii) E =`0∪`1 with `0, `1 defined over GF(q),`0 k`1; Ω∩ U = Ω0∪Ω1,|Ω0∩Ω1|= 1 p>2;
(iii) E =`0∪`1 with `0, `1 defined over GF(q),`0 =`1; Ω∩ U is a classical Baer suboval ofΩ, p>2;
(iv) E =`0∪`1 with `0, `1 defined over GF(q2),|`0∩`1|= 1; Ω∩ U ={P∞,A};
(v) E =`0∪`1 with `0, `1 defined over GF(q2),`0k`1; Ω∩ U ={P∞}; p >2.
Proof Classification Ω ∩ U
E is an absolutely reducible conic
Proposition
IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and 2,2q for p= 2. More precisely, one of the following cases occurs:
(i) E =`0∪`1 with `0, `1, defined over GF(q),|`0∩`1|= 1;
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely reducible conic
Proposition
IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and 2,2q for p= 2. More precisely, one of the following cases occurs:
(i) E =`0∪`1 with `0, `1, defined over GF(q),|`0∩`1|= 1;
Ω∩ U = Ω0∪Ω1,|Ω0∩Ω1|= 2;
(ii) E =`0∪`1 with `0, `1 defined over GF(q),`0 k`1; Ω∩ U = Ω0∪Ω1,|Ω0∩Ω1|= 1 p>2;
(iii) E =`0∪`1 with `0, `1 defined over GF(q),`0 =`1; Ω∩ U is a classical Baer suboval ofΩ, p>2;
(iv) E =`0∪`1 with `0, `1 defined over GF(q2),|`0∩`1|= 1;
(v) E =`0∪`1 with `0, `1 defined over GF(q2),`0k`1; Ω∩ U ={P∞}; p >2.
Proof Classification Ω ∩ U
E is an absolutely reducible conic
Proposition
IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and 2,2q for p= 2. More precisely, one of the following cases occurs:
(i) E =`0∪`1 with `0, `1, defined over GF(q),|`0∩`1|= 1;
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely irreducible conic
a23=b23= 0 Proposition
IfE is an irreducible conic, then
|U ∩Ω|=
q,q+ 1,q+ 2 when p >2; q,q+ 2 when p = 2.
Proof Classification Ω ∩ U
E is an absolutely irreducible conic
a23=b23= 0
Proposition
IfE is an irreducible conic, then
|U ∩Ω|=
q,q+ 1,q+ 2 when p >2; q,q+ 2 when p = 2.
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely irreducible conic
a23=b23= 0 Proposition
IfE is an irreducible conic, then
|U ∩Ω|=
q,q+ 1,q+ 2 when p >2;
q,q+ 2 when p = 2.
Proof Classification Ω ∩ U
E is an absolutely reducible cubic
either a236= 0 or b236= 0. Proposition
IfE is an absolutely reducible cubic, then E=`∪ D where` is a line andDis an absolutely irreducible conic, both defined over GF(q).
(i) U ∩Ω = Ω0∪Ω1 with Ω0∩Ω1 =∅ when `external to D; (ii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 1 when `tangent toD
and p>2;
(iii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 2 when `secant toD.
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely reducible cubic
either a236= 0 or b236= 0.
Proposition
IfE is an absolutely reducible cubic, then E=`∪ D where` is a line andDis an absolutely irreducible conic, both defined over GF(q).
(i) U ∩Ω = Ω0∪Ω1 with Ω0∩Ω1 =∅ when `external to D; (ii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 1 when `tangent toD
and p>2;
(iii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 2 when `secant toD.
Proof Classification Ω ∩ U
E is an absolutely reducible cubic
either a236= 0 or b236= 0.
Proposition
IfE is an absolutely reducible cubic, then E=`∪ D where` is a line andDis an absolutely irreducible conic, both defined over GF(q).
(i) U ∩Ω = Ω0∪Ω1 with Ω0∩Ω1 =∅ when `external to D; (ii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 1 when `tangent toD
and p>2;
(iii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 2 when `secant toD.
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely reducible cubic
either a236= 0 or b236= 0.
Proposition
IfE is an absolutely reducible cubic, then E=`∪ D where` is a line andDis an absolutely irreducible conic, both defined over GF(q).
(i) U ∩Ω = Ω0∪Ω1 with Ω0∩Ω1 =∅ when `external to D;
(ii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 1 when `tangent toD and p>2;
(iii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 2 when `secant toD.
Proof Classification Ω ∩ U
E is an absolutely reducible cubic
either a236= 0 or b236= 0.
Proposition
IfE is an absolutely reducible cubic, then E=`∪ D where` is a line andDis an absolutely irreducible conic, both defined over GF(q).
(i) U ∩Ω = Ω0∪Ω1 with Ω0∩Ω1 =∅ when `external to D;
(ii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 1 when `tangent toD and p>2;
(iii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 2 when `secant toD.
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely reducible cubic
either a236= 0 or b236= 0.
Proposition
IfE is an absolutely reducible cubic, then E=`∪ D where` is a line andDis an absolutely irreducible conic, both defined over GF(q).
(i) U ∩Ω = Ω0∪Ω1 with Ω0∩Ω1 =∅ when `external to D;
(ii) U ∩Ω = Ω0∪Ω1 with |Ω0∩Ω1|= 1 when `tangent toD and p>2;
Proof Classification Ω ∩ U
E is an absolutely irreducible cubic
Proposition
IfE is an absolutely irreducible cubic, then
|Ω∩ U | ∈ [(√
q−1)2,(√
q+ 1)2]. Remark
The method in the above proof may work when eitherΩis a translation oval or Segre’s oval orU is a Buekenhout-Metz unital. However the arising plane curveE may not have low degree. Remark
The method in the above proof may also work in PG(r,q2) with r≥3when Ωis a rational normal curve and U is a Hermitian variety.
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely irreducible cubic
Proposition
IfE is an absolutely irreducible cubic, then
|Ω∩ U | ∈ [(√
q−1)2,(√
q+ 1)2].
Remark
The method in the above proof may work when eitherΩis a translation oval or Segre’s oval orU is a Buekenhout-Metz unital. However the arising plane curveE may not have low degree. Remark
The method in the above proof may also work in PG(r,q2) with r≥3when Ωis a rational normal curve and U is a Hermitian variety.
Proof Classification Ω ∩ U
E is an absolutely irreducible cubic
Proposition
IfE is an absolutely irreducible cubic, then
|Ω∩ U | ∈ [(√
q−1)2,(√
q+ 1)2].
Remark
The method in the above proof may work when eitherΩis a translation oval or Segre’s oval orU is a Buekenhout-Metz unital.
However the arising plane curveE may not have low degree.
Remark
The method in the above proof may also work in PG(r,q2) with r≥3when Ωis a rational normal curve and U is a Hermitian variety.
G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes
Proof Classification Ω ∩ U
E is an absolutely irreducible cubic
Proposition
IfE is an absolutely irreducible cubic, then
|Ω∩ U | ∈ [(√
q−1)2,(√
q+ 1)2].
Remark
The method in the above proof may work when eitherΩis a translation oval or Segre’s oval orU is a Buekenhout-Metz unital.
However the arising plane curveE may not have low degree.
Remark
2