# Projekt:Computeralgebra-Berechnungen/Symmetrische Hilbert-Kunz Theorie/Fermat-Flächen im Vergleich

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Betrachte die MKSC des maximalen Ideals auf einer Fermat-Fläche. Beobachtung: Für kleine q hängt das nicht vom Grad der Fläche ab. Allerdings unterscheiden sich die Summanden umso mehr, je größer q ist.

q=1 ${\displaystyle +\sum _{m}h^{1}(Syz_{1}(m))}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
MKSC ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle 1}$
q=2 ${\displaystyle +\sum _{m}h^{1}(S^{2}(Syz_{1})(m))}$ ${\displaystyle 10}$ ${\displaystyle 10}$ ${\displaystyle 10}$
${\displaystyle -\sum _{m}h^{1}({\mathcal {T}}_{2}^{(0)}(m))}$ ${\displaystyle -4}$ ${\displaystyle -4}$ ${\displaystyle -4}$
${\displaystyle +\sum _{m}h^{1}({\mathcal {T}}_{2}^{(1)}(m))}$ ${\displaystyle 2}$ ${\displaystyle 2}$ ${\displaystyle 2}$
MKSC ${\displaystyle 8}$ ${\displaystyle 8}$ ${\displaystyle 8}$
q=3 ${\displaystyle +\sum _{m}h^{1}(S^{3}(Syz_{1})(m))}$ ${\displaystyle 40}$ ${\displaystyle 50}$ ${\displaystyle 50}$
${\displaystyle -\sum _{m}h^{1}({\mathcal {T}}_{3}^{(0)}(m))}$ ${\displaystyle -20}$ ${\displaystyle -30}$ ${\displaystyle -30}$
${\displaystyle +\sum _{m}h^{1}({\mathcal {T}}_{3}^{(1)}(m))}$ ${\displaystyle 14}$ ${\displaystyle 14}$ ${\displaystyle 14}$
MKSC ${\displaystyle 34}$ ${\displaystyle 34}$ ${\displaystyle 34}$
q=4 ${\displaystyle +\sum _{m}h^{1}(S^{4}(Syz_{1})(m))}$ ${\displaystyle 119}$ ${\displaystyle 156}$ ${\displaystyle 175}$
${\displaystyle -\sum _{m}h^{1}({\mathcal {T}}_{4}^{(0)}(m))}$ ${\displaystyle -69}$ ${\displaystyle -106}$ ${\displaystyle -125}$
${\displaystyle +\sum _{m}h^{1}({\mathcal {T}}_{4}^{(1)}(m))}$ ${\displaystyle 55}$ ${\displaystyle 55}$ ${\displaystyle 55}$
${\displaystyle -\sum _{m}h^{1}({\mathcal {T}}_{4}^{(2)}(m))}$ ${\displaystyle -1}$ ${\displaystyle -1}$ ${\displaystyle -1}$
MKSC ${\displaystyle 104}$ ${\displaystyle 104}$ ${\displaystyle 104}$
q=5 ${\displaystyle +\sum _{m}h^{1}(S^{5}(Syz_{1})(m))}$ ${\displaystyle 284}$ ${\displaystyle 377}$ ${\displaystyle 456}$
${\displaystyle -\sum _{m}h^{1}({\mathcal {T}}_{5}^{(0)}(m))}$ ${\displaystyle -179}$ ${\displaystyle -272}$ ${\displaystyle -351}$
${\displaystyle +\sum _{m}h^{1}({\mathcal {T}}_{5}^{(1)}(m))}$ ${\displaystyle 161}$ ${\displaystyle 161}$ ${\displaystyle 161}$
${\displaystyle -\sum _{m}h^{1}({\mathcal {T}}_{5}^{(2)}(m))}$ ${\displaystyle -7}$ ${\displaystyle -7}$ ${\displaystyle -7}$
MKSC ${\displaystyle 259}$ ${\displaystyle 259}$ ${\displaystyle 259}$