The minimal dimension of a faithful linear representation of all finite subgroups of the two-dimensional Cremona group is 8 when the field contains a primitive cube root of unity, 6 otherwise, and infinite in positive characteristic.
Forms of del Pezzo surfaces of degree 5 and 6
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In this paper we obtain necessary and sufficient condition for existence of del Pezzo surfaces of degree $5$ and $6$ over a field $K$ with a prescribed action of absolute Galois group $\text{Gal} ( K^{\text{sep}}/K)$ on the graph of $(-1)$-curves. Also we compute automorphism groups of del Pezzo surfaces of degree $5$ over arbitrary fields.
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Representations of finite subgroups of Cremona groups
The minimal dimension of a faithful linear representation of all finite subgroups of the two-dimensional Cremona group is 8 when the field contains a primitive cube root of unity, 6 otherwise, and infinite in positive characteristic.