We use non-reductive geometric invariant theory (NRGIT) to study a moduli problem involving pure sheaves of fixed Harder-Narasimhan type. The construction of moduli spaces for such sheaves can be reduced to the construction of NRGIT quotients of subschemes of Quot schemes. Hitherto NRGIT has been developed only for actions on varieties, whereas Quot schemes are not varieties.

So first we put the NRGIT developed by Bérczi et al ([Bér+18b], [Bér+16]) into a scheme-theoretic framework. Linear actions of ˆU := U ⋊λ Gm are considered, where U is a unipotent group graded by λ : Gm → Aut(U ). The condition “semistability coincides with stability” (or “ss=s”) required by Bérczi et al is enhanced scheme-theoretically for an ample ˆU -linearisation on a projective scheme X. With “ss=s”, we show that an open subscheme of X has a geometric quotient X// ˆU , which is not necessarily reduced. When “ss=s” fails but a weaker condition holds, a blow-up of X satisfies “ss=s”. For reduced X, our results are close to those of Bérczi et al, except we need only one blow-up.

Next we apply our NRGIT to the moduli problem. Besides the Harder- Narasimhan type τ , we fix the vector δ representing dimensions of spaces of unipotent endomorphisms. We extend the notion of (τ, δ)-stable sheaves considered by Hoskins and Jackson ([Jac21], [HJ21]) to (τ, δ)-stable families, and define a moduli functor M′τ,δ classifying (τ, δ)-families. A coarse moduli space for M′τ,δ can be constructed as a NRGIT quotient. Hoskins and Jackson considered the reduced version of this NRGIT problem. Now we study it scheme-theoretically. When τ has length 2, we can perform the blow-up and apply our NRGIT. For general lengths, we need additional hypotheses on τ to satisfy the condition for blow-ups.