What is the moduli functor $\mathbb{P}Ext^1(L,M)$ represents

Question

Let $X$ be an algebraic space and $L,M$ are vector bundles with rank $n,m$. Then, It is known that $\mathbb{P}Ext^1(L,M)$ is a parameter space for isomorphism classes of vector bundle of rank $n+m$, which is obtained by extension of $L$ by $M$.

My Question is, What is moduli functor $\mathbb{P}Ext^1(L,M)$ represents?.

In the book of Huybrechts & Lehn, there is a familly, which they call "universal familly" $\mathcal{U}$ on X $\times$ $\mathbb{P}Ext^1(L,M)$, that is

$0 \to (\pi_1)^*M\otimes(\pi_2)^*\mathcal{O}(1) \to \mathcal{U} \to (\pi_1)^*L \to 0$.

So, my second question is that is this really a universal family for moduli functor that $\mathbb{P}Ext^1(L,M)$?.

Thank you for read my questions.

Answer 1

You answered your own question. Let $F:\text{Alg. Sp.} \to \text{Sets}$ be the functor that associates to every algebraic space $T$ the set of equivalence classes of triples $(\mathcal{L},\mathcal{V},q,p)$ of an invertible sheaf $\mathcal{L}$ on $T$, a locally free sheaf $\mathcal{V}$ on $X\times T$, a homomorphism of coherent sheaves on $X\times T$, $$ q:\text{pr}_X^*M\otimes_{\mathcal{O}_{X\times T}} \text{pr}_T^*\mathcal{L} \to \mathcal{V},$$ and a homomorphism of coherent sheaves on $X\times T$, $$ p:\mathcal{V}\to \text{pr}_X^* L,$$ such that the following sequence is a short exact sequence, $$ 0 \to \text{pr}_X^*M\otimes_{\mathcal{O}_{X\times T}}\text{pr}_T^*\mathcal{L} \to \mathcal{V} \to \text{pr}_X^*L \to 0. $$ An equivalence from the triple $(\mathcal{L},\mathcal{V},q,p)$ to a second equivalence $(\mathcal{L}',\mathcal{V}',q',p')$ is a pair $(\alpha,\beta)$ of an isomorphism $\alpha:\mathcal{L}\to \mathcal{L}'$ of invertible sheaves on $T$ and an isomorphism $\beta:\mathcal{V}\to \mathcal{V}'$ of locally free sheaves on $X\times T$ such that both $p'\circ \beta$ equals $p$ and such that $\beta\circ q$ equals $q'\circ \text{Id}_M\otimes \alpha$.

The functor of equivalence classes of triples is represented by the universal object $\mathcal{U}$.

Edit. I forgot to exclude the trivial extension. We need to add the hypothesis that for every geometric point of $T$, the corresponding extension of $L$ by $M$ is nontrivial.