I study Nitin Nitsure's paper *Construction of Hilbert and Quot Schemes* (arXiv:math/0504590) and have some problems with the content of imposed universal property **(F)** in the section "Use of Flattening Stratification" (p 26).

Reading the imposed universal property **(F)** following parts confuse me:

**Question 1:** About the composition map $h$. First of all I guess the author
had $W_Y \otimes_{O_Y} \operatorname{Sym}^r V_Y$ in mind and
not $W \otimes_{O_S} \operatorname{Sym}^r V$, or not?

Let assume that. But why is
$\pi_Y^* W_Y \otimes_{O_Y} \operatorname{Sym}^r V_Y
= \pi_Y^* \pi_{Y*} E_Y$ and **not** $\pi_Y^* \pi_{Y*} E_Y(r)$?

I think that this identification should arise in same way as in the construction of

$\alpha:$ $ \frak{Quot} $ $^{\Phi, L}_{E/X/S} \to $ $\frak{Grass} $ $(W \otimes_{O_S} \operatorname{Sym}^r V, \Phi(r))$

from the section "Embedding Quot into Grassmannian" (the previous part on same page). Let introduce abbreviations for these functors:

We set $G:=$ $\frak{Grass}$ $(W \otimes_{O_S} \operatorname{Sym}^r V, \Phi(r))$ and $Q:=$ $\frak{Quot}$ $^{\Phi, L}_{E/X/S}$. The explicit construction of $\alpha$ is given there.

I assume that the author hasn't changed notations when passing to "Use of Flattening Stratification" and therefore it should still be $\pi_Y^* W_Y \otimes_{O_Y} \operatorname{Sym}^r V_Y = \pi_Y^* \pi_{Y*} E_Y(r)$?

**Question 2:** Based on insights from Question 1 I think that
the cokernel of $h$ should be $q: E_Y(r) \to F$ because
$h$ should be $\pi_Y^* K_Y \to E_Y(r)$.
Then I come to conclusion that the universal property **(F)** on $F$ should be read as follows:

(UP)$F(-r)$ (ie twist of $F$ by $O_Y(-r)$ and not $F$ itself !) is flat over $Y$ with its Hilbert polynomial on all fibers equal to $\Phi$ if and only if $\phi : Y \to T$ factors via $T' \to T$.

But then the proof shows an embedding of $ \frak{Quot} $ $^{\Psi, L}_{E(-r)/X/S} $ with $\Psi(X) := \Phi(X-r)$ and **not** $ \frak{Quot} $ $^{\Phi, L}_{E/X/S} $. That's a problem.

The reason I think so is the definition of relative representability, see Tag 0023 from the Stacks Project.

Reparaphrasing this we have to show for fixed $f \in G(Y) \cong \operatorname{Hom}(Y, \operatorname{Grass}(W \otimes_{O_S} \operatorname{Sym}^r V, \Phi(r)))$ that $h_T \times_{f, G, \alpha} Q(Y)$ is the set of pairs $(\phi, q) \in h_T(Y) \times Q(Y)$ where $q: E_Y \to D$ is the quotient with $G(\phi)(f)=\alpha(q)$.

In this sense we have to show that this is representable. Now what does this condition mean? By construction $ G(\phi)(f)= f_Y: W_Y \otimes_{O_Y} \operatorname{Sym}^r V_Y \to \phi^* \mathcal{J}$ and for a quotient $q: E_Y \to D \in Q(Y)$ the image $\alpha(q)$ is by construction of $\alpha$ exactly $\pi_{Y*} E_Y(r) \to \pi_{Y*} D(r) $ and the requirement is that $\alpha(q)= f_Y$.

Therefore I think that the correct formulation in the last
sentence of universal **(F)** should be as in **(UP)**. Is that true or do I make a thinking error and misunderstand the construction?

**Question 3:** I not understand why if we take at the end
$T:= \operatorname{Grass}(W \otimes_{O_S} \operatorname{Sym}^r, \Phi(r))$ then the corresponding locally closed subscheme $T' \subset T$
represents the functor $\frak{Quot}$ $^{\Phi, L}_{E/X/S}$
"by construction". Precisely, by the construction of what?
Of $\alpha$?