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In this paper, we introduce the notion of the diagonal property and the weakpoint property for an ind-variety. We prove that the ind-varieties of higherrank divisors of integral slopes on a smooth projective curve have the weakpoint property. Moreover, we show that the ind-variety of $(1,n)$-divisors hasthe diagonal property. Furthermore, we obtain that the Hilbert schemesassociated to the good partitions of a constant polynomial satisfy the diagonalproperty. On the process of obtaining this, we provide an upper bound on thenumber of such Hilbert schemes up to isomorphism. Furthermore, we prove thatthe obtained upper bound is attained in case of genus zero curves and henceconclude that the bound is sharp.

Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes