Before rigorously proving this, let us explain how to arrive to Theorem 3.1 and see it just by looking at the geometric data involved. What follows in not completely rigorous but provides, in our opinion, a nice geometric picture.

Blowing up 𝔹 along 𝔹′ substitutes 𝔹′ by the projective normal bundle of 𝔹′. So a point of 𝔹′ represented by a line *L* ∈ $\begin{array}{}{\mathbb{P}}_{2}^{\ast}\end{array}$ and a point *p* ∈ *L*, which is encoded by some 〈*w*〉 ∈ ℙ*H*^{0}(L, 𝓞_{L}(1)), is substituted by the projective space *D*_{(L,p)} of the normal space T_{(L,p)} 𝔹/T_{(L,p)}𝔹′ to 𝔹′ at (*L*, *p*).

As 𝔹 is a projective bundle over *N*, and 𝔹′ is a ℙ_{1}-bundle over *N*′, the normal space is a direct sum of the normal spaces along the base and along the fibre. Therefore, *D*_{(L,p)} is the join of the corresponding projective spaces: of ℙ_{3} = *L*^{[3]} (normal projective space to *N*′ in *N* at *L* ∈ *N*′) and ℙ_{9} = ℙ(*S*^{3} *V*^{*}) (normal projective space to *L*^{*} in *J*(*L*^{*}, ℙ(*S*^{3} *V*^{*})) at *p* ∈ *L* ≅ *L*^{*}; notice that the normal projective bundle of *L*^{*} ⊆ *J*(*L*^{*}, ℙ(*S*^{3} *V*^{*})), i. e., ℙ_{1} ⊆ ℙ_{11}, is trivial).

The fibre *J*(*L*^{*}, ℙ(*S*^{3} *V*^{*})) of 𝔹 → *N* over *L* ∈ *N*′ is substituted under the blow-up by the fibre that consists of two components: the first component is the blow-up of *J*(*L*^{*}, ℙ(*S*^{3} *V*^{*})) along *L*^{*}, the second one is a projective bundle over *L*^{*} with the fibre ℙ_{13} = *J*( *L*^{[3]}, ℙ(*S*^{3} *V*^{*}) ), the components intersect along *L*^{*} × ℙ(*S*^{3} *V*^{*}).

Fig. 3 The fibre of 𝔹͠ over *L* ∈ *N*′.

The space *L*^{[3]} is naturally identified with the projective space of cubic forms on *L*^{*} whereas ℙ(*S*^{3} *V*^{*}) is clearly the space of cubic curves on ℙ_{2}.

Assume *L* = *Z*(*x*_{0}) such that {*x*_{0}, *x*_{1}, *x*_{2}} is a basis of *V*^{*} = *H*^{0}(ℙ_{2}, 𝓞_{ℙ2}(1)). Identifying *x*_{1} and *x*_{2} with their images in *H*^{0}(*L*, 𝓞_{L}(1)), {*x*_{1}, *x*_{2}} is a basis of *H*^{0}(*L*, 𝓞_{L}(1)).

We conclude that the join of *L*^{[3]} and ℙ *S*^{3} *V*^{*} can be identified with the projective space corresponding to the vector space

$$\begin{array}{}{\displaystyle \{\lambda \cdot {x}_{0}h+{\lambda}^{\prime}\cdot wg\mid h({x}_{0},{x}_{1},{x}_{2})\in {S}^{3}{V}^{\ast},g({x}_{1},{x}_{2})\in {H}^{0}(L,{\mathcal{O}}_{L}(3)),(\lambda ,{\lambda}^{\prime}){\in}^{2}\},}\end{array}$$

i. e., the space of planar quartic curves through the point *p* = *Z*(*x*_{0}, *w*).

So the exceptional divisor of the blow-up Bl_{𝔹′}𝔹 is a projective bundle with fibre over (*L*, *p*) being interpreted as the space of quartic curves through *p*. This way we obtain a map from the exceptional divisor to the universal quartic *M*_{1}. Its fibre over a pair *p* ∈ *C* is identified with the space of lines *L* ∈ $\begin{array}{}{\mathbb{P}}_{2}^{\ast}\end{array}$ through *p*, i. e., with a projective line. Contracting the exceptional divisor along these lines one gets *M*. The contraction is possible by [15, 16, 17], which can be seen as follows.

The fibre of *D* → *M*_{1} over a pair *p* ∈ *C* may be identified with the fibre $\begin{array}{}{\mathbb{B}}_{p}^{\mathrm{\prime}}\end{array}$ over *p* ∈ ℙ_{2} of the map 𝔹′ → ℙ_{2} given by the projection to the second factor (cf. (7). Every two points (*L*, *p*) and (*L*′, *p*) of $\begin{array}{}{\mathbb{B}}_{p}^{\mathrm{\prime}}\end{array}$ ⊆ 𝔹′ are substituted by the projective spaces *J*(
*L*^{[3]}, ℙ(*S*^{3} *V*^{*})) and *J*(*L*′^{[3]}, ℙ(*S*^{3} *V*^{*})) respectively, each of which is naturally identified with the space of quartics through *p*. Assume without loss of generality *p* = 〈0, 0, 1〉.

The fibre $\begin{array}{}{\mathbb{B}}_{p}^{\mathrm{\prime}}\end{array}$ in this case is identified with the space of lines in ℙ_{2} through *p*, i. e., with the projective line in *N*′ = $\begin{array}{}{\mathbb{P}}_{2}^{\ast}\end{array}$ = ℙ*V*^{*} consisting of classes of linear forms *α* *x*_{0} + *β* *x*_{1}, 〈*α*, *β*〉 ∈ ℙ_{1}. The fibre has a standard covering $\begin{array}{}{\mathbb{B}}_{p,0}^{\mathrm{\prime}}\end{array}$ = {*x*_{0} + *β* *x*_{1}}, $\begin{array}{}{\mathbb{B}}_{p,1}^{\mathrm{\prime}}\end{array}$ = {*α* *x*_{0} + *x*_{1}}, which is induced by the standard covering of $\begin{array}{}{\mathbb{P}}_{2}^{\ast}\end{array}$. The elements of the fibre corresponding to the points of $\begin{array}{}{\mathbb{B}}_{p,0}^{\mathrm{\prime}}\end{array}$ are the equivalence classes of matrices

$$\begin{array}{}{\displaystyle {A}_{0}=\left(\begin{array}{ccc}0& {x}_{1}\cdot {x}_{2}& -{x}_{1}\cdot {x}_{1}\\ -{x}_{2}& 0& {x}_{0}+\beta {x}_{1}\\ {x}_{1}& -({x}_{0}+\beta {x}_{1})& 0\end{array}\right).}\end{array}$$

The elements of the fibre corresponding to the points of $\begin{array}{}{\mathbb{B}}_{p,1}^{\mathrm{\prime}}\end{array}$ are the equivalence classes of matrices

$$\begin{array}{}{\displaystyle {A}_{1}=\left(\begin{array}{ccc}0& {x}_{0}\cdot {x}_{2}& -{x}_{0}\cdot {x}_{0}\\ -{x}_{2}& 0& \alpha {x}_{0}+{x}_{1}\\ {x}_{0}& -(\alpha {x}_{0}+{x}_{1})& 0\end{array}\right).}\end{array}$$

In this way, we have chosen, so to say, the normal forms for the representatives of the $\begin{array}{}{\mathbb{B}}_{p}^{\mathrm{\prime}}\end{array}$. For *β* = *α*^{−1}, i. e., on the intersection of $\begin{array}{}{\mathbb{B}}_{p,0}^{\mathrm{\prime}}\end{array}$ and $\begin{array}{}{\mathbb{B}}_{p,1}^{\mathrm{\prime}}\end{array}$, these matrices are equivalent. One computes that (*g* *A*_{1} *h* = *A*_{0}) for matrices

$$\begin{array}{}{\displaystyle g=\left(\begin{array}{ccc}{\alpha}^{3}& {\alpha}^{2}{x}_{0}-\alpha {x}_{1}& {\alpha}^{2}{x}_{2}\\ 0& 1& 0\\ 0& 0& -\alpha \end{array}\right),\phantom{\rule{1em}{0ex}}h=\left(\begin{array}{ccc}1& 0& 0\\ {\alpha}^{-1}& -{\alpha}^{-2}& 0\\ 0& 0& {\alpha}^{-1}\end{array}\right)}\end{array}$$

with determinants

$$\begin{array}{}{\displaystyle detg=-{\alpha}^{4},\phantom{\rule{1em}{0ex}}deth=-{\alpha}^{-3}.}\end{array}$$

Consider the automorphism $\begin{array}{}{\mathbb{W}}^{s}\stackrel{\xi}{\to}{\mathbb{W}}^{s},A\mapsto \end{array}$ *gAh*. Then

$$\begin{array}{}{\displaystyle det(\xi ({A}_{1}+{B}_{1}))=\alpha \cdot det({A}_{1}+{B}_{1}).}\end{array}$$(11)

From (11) it follows that the restriction of the ideal sheaf of *D* to a fibre of the morphisms *D* → *M*_{1} is 𝓞_{ℙ1}(1). By [15, 16, 17], this means that one can blow down *D* in 𝔹͠ along the map *D* → *M*_{1}. This gives the blow down Bl_{𝔹′}𝔹 → *M* that contracts the exceptional divisor of Bl_{𝔹′}𝔹 along all lines $\begin{array}{}{\mathbb{B}}_{p}^{\mathrm{\prime}}\end{array}$.

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