Average size of the automorphism group of smooth projective hypersurfaces over finite fields

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Abstract: In this paper, we show that the average size of the automorphism group over

m a t h b b F_{q}

of a smooth degree

d

hypersurface in

m a t h b b P_{m a t h b b F_{q}}^{n}

is equal to

1

as

d i g h t a r r o w i n f t y

. We also discuss some consequence of this result for the moduli space of smooth degree

d

hypersurfaces in

m a t h b b P^{n}

.

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