\begin{equation}
\label{eq:asymp-exp-B'}
B'(y) = B'(\rho_B) - B''(\rho_B)(\rho_B-y) + \bconst (\rho_B-y)^{3/2} + O\((\rho_B-y)^2\)
\end{equation}

\begin{theorem}\label{thm:phasetransition_block_sizes} Let $u>0$. The block-weighted graph $\Gnu$ exhibits the following behaviours when $n\to\infty$.
\begin{description}
    \item[Subcritical case.] If $u<u_C$, then \cref{th-taille-blocs-sous-crit}  holds for block sizes:
    \begin{align*}
    \frac{1}{(c(u) n)^{2/3}}&\(\(1-\frac{u}{u_C}\)n - \LB{1}{\Gnu}, \(\LB{j}{\Gnu} : j\geq 2\)\) \\
    &\hspace{170pt} \xrightarrow[n\to\infty]{(d)} \(L_1,\(\Delta L_{(j-1)} : j\geq 2\)\)
    \end{align*}
    where $(L_t)_{t\in[0,1]}$ is a stable process of parameter $3/2$ such that $\E{e^{-sL_1}} = e^{\Gamma(-3/2)s^{3/2}}$ and $\Delta L_{(1)} \geq \Delta L_{(2)} \geq \dots$ is the ranked sequence of its jumps.

    \item[Critical case.] If $u=u_C$, then \cref{eq:size-bloc-critical} holds for the block sizes:
    \[\(\frac{\LB{j}{\G_{n,u_C}}}{n^{2/3}} : j \geq 1\) \xrightarrow[n\to\infty]{(d)} \(E_{(j)} : j \geq 1\),\]
    where the $E_{(j)}$ are as defined in \cref{prop:degrees-Tnu}.
    \item[Supercritical case.] If $u>u_C$, then \cref{eq:grandbloc} holds for the block sizes (fixing $j \in \mathbb{N}_0$):
    \[\LB{j}{\Gnu} = \frac{\ln(n)}{\ln\(\frac{\rho_B}{y(u)}\)} - \frac{5}{2}\frac{\ln(\ln(n))}{\ln\(\frac{\rho_B}{y(u)}\)} + O_{\mathbb{P}}(1).\]
\end{description}
\end{theorem}

\begin{theorem}\label{thm:phase-transition}
Let $\cC$ be a block-stable class of connected graphs, let $\cB$ be its class of $2$-connected components whose EGF we denote by $B(y)$, and let $\rho_B$ be the radius of convergence of $B$. Let
\[u_C := \frac{1}{\rho_B B''(\rho_B)}.\]
Then the following hold.
\begin{itemize}
    \item The singular behaviour of $C^{\bullet}(x,u)$ depends on the relative positions of $u$ and $u_C$;
    \item The block tree corresponding to a graph drawn according to $\Pnu$ follows a subcritical \BGW{} tree law if $u<u_C$, and a critical \BGW{} tree law if $u \ge u_C$.
\end{itemize}
\end{theorem}

\begin{theorem}
\label{th:dvp-sing-Cbxu}
The series $C^\bullet(x,u)$ displays the following asymptotic behaviours when $x$ is in a $\Delta$-domain neighbourhood of $\rho(u)$, where we set $X := 1-\frac{x}{\rho(u)}$.
\begin{description}
    \item[Subcritical regime.] If $u<u_C$, then
   \[C^\bullet(x, u) = \rho_B - r(u)X  + s(u) X^{3/2} + \Theta\(X^2\),\]
   where
   \[r(u) := \rho_B\(1-\frac{u}{u_C}\)^{-1}\qquad\text{and}\qquad s(u) := u \bconst \rho_B^{5/2} \(1-\frac{u}{u_C}\)^{-5/2}.\]
   Thus,
   \[[x^n] C^\bullet(x, u) \sim \frac{3 s(u)}{4 \sqrt{\pi}} n^{-5/2} \rho(u)^{-n}.\]
    \item[Critical regime.] If $u = u_C$, then
    \[C^\bullet(x, u_C) = \rho_B - s X^{2/3} + \Theta\(X\),\]
    where $s := u_C^{-2/3} \bconst^{-2/3}$. Thus,
    \[[x^n] C^\bullet(x, u_C) \sim \frac{s \Gamma(2/3)}{\sqrt{3}\pi} n^{-5/3} \rho(u_C)^{-n}.\]
     \item[Supercritical regime.] If $u>u_C$, then
    \[C^\bullet(x, u) = y(u) - s(u) X^{1/2} + \Theta\(X\),\]
    where
    \[s(u):=\sqrt{\frac{2}{uB''(y(u)) + u^2 B'(y(u))^2}} = \sqrt{\frac{2y(u)}{1+ y(u)\ln\(\frac{y(u)}{\rho(u)}\)^{2}}}.\]
    Thus,
    \[[x^n] C^\bullet(x, u) \sim \frac{s(u)}{2\sqrt{\pi}} n^{-3/2} \rho(u)^{-n}.\]
\end{description}
\end{theorem}

\begin{enumerate}
    \item \emph{Block graph} $B(\g)$ (see e.g., \cite{harary1969}): The vertices of $B(\g)$ are the blocks of $\g$, and two blocks are joined by an edge in $B(\g)$ if they overlap at a vertex. The resulting graph $B(\g)$ is characterised by the property of being connected and that all of its blocks are cliques.
    \item \emph{Block-cut graph} $T(\g)$ (see e.g., \cite{diestel2025graph}): The vertices of $T(\g)$ are the blocks and the cut-vertices of $\g$, and a cut-vertex is joined to a block by an edge in $T(\g)$ if that vertex is contained in that block. The resulting graph $T(\g)$ is a tree.
\end{enumerate}

In the present work, we use an alternative block decomposition that naturally arises from the Lagrangean form of \eqref{eq:Cbullet-block} of the generating series $C^{\bullet}(x)$. It originates from work of Addario-Berry \cite{2Louigi} on planar maps (see also \cite{Sal24}), and we give a precise algorithmic description in \cref{subsec:anal-comb-block-tree}. We note that this construction works specifically for \emph{rooted} labelled connected graphs, and we call its output a \emph{decorated block tree}. In a decorated block tree, the vertices represent all the vertices of the original graph $\g$ but each vertex is decorated by a set of blocks that contain that vertex. In \cite{2Louigi}, the decorations were simply blocks, but in our construction the decorations are more general \emph{sets of blocks}.

\section{Decorated block trees of connected block-stable graphs}
\label{sec:decorated}
In this section, we describe \emph{decorated block trees}, which are a central tool in our analysis.

\subsection{Construction of decorated block tree}
\label{subsec:anal-comb-block-tree}
Recalling the Lagrangean form of \cref{eq:Cbullet-block}, we define
\begin{equation}
\label{eq:def-Phi}
\Phi(y) = \sum_{k=0}^\infty \frac{\phi_k}{k!} y^k \coloneqq \exp(B'(y)) \in \mathbb C [[y]],
\end{equation}
which is precisely the exponential generating function for the class $\Set \circ \mathcal{B}'$. For convenience we also use $\Phi$ to denote  the class $\Set \circ \mathcal{B}'$. It follows from \cref{eq:Cbullet-block} that
\[C^\bullet(x) = x \Phi(C^\bullet(x)).\]
As explained in \cite[\S 2.2.1]{Sal24}, there is a systematic way to define a (plane) block tree from a Lagrangean decomposition scheme. Namely, each vertex of the tree is decorated with an object $\varphi$ from the class $\Phi$, and the subtrees attached to a vertex of the tree (corresponding to some $\varphi$) are the trees of the rooted graphs substituted into $\varphi$. The situation in the case of graphs is a little more involved than in the case of maps because of the labelling possibilities. Thus, we use the following notion.

Remember that elements of the class $\Phi$ are well-labelled. Therefore, for each $\g \in \cC$, when we consider sets of derived blocks of $\g$, it is always implied that the set has been relabelled with a well-labelling which is consistent with the order of the labels in $\g$.

\begin{figure}
\begin{center}
\includegraphics[width=1\textwidth]{images/block-tree.pdf}
\end{center}
\caption{The image on the left is a connected labelled rooted planar graph $\g$, with root highlighted in grey. The image on the right is the associated decorated block tree $\TT_\g$.}
\label{fig:block-tree-2}
\end{figure}

Now, we give an algorithmic definition of decorated block trees. An example is provided in \cref{fig:block-tree-2}.

\begin{definition}
For $\g\in \cC^\bullet$ (which for ease of presentation $\g$ is also allowed to be a rooted derived graph, with a root and one unlabelled non-root vertex), we define the \emph{decorated block tree $\TT_\g$} as the result of the following construction:
\begin{enumerate}
    \item Let $r$ be the root of $\g$, and let $\varphi = \{\b'_1, \ldots, \b'_\ell\}$ be the set of derived blocks containing $r$ \emph{and no unlabelled vertex} in $\g$, where the label of the vertex corresponding to $r$ is removed. Start with a root vertex $r_{\TT}$ in $\TT_\g$ and decorate it with $\varphi$.
    \item For $i \in \{1, \ldots, \ell\}$, we construct a derived graph $\g'_{\b_i}$ by deleting $r$ from $\g$, keeping the connected component of $\g$ that intersects $\b'_i$, and then adding $r$ back to this subgraph. We consistently relabel the vertices of $\g'_{\b_i}$ except $r$ which remains unlabelled. For every labelled vertex $v$ of $\b'_i$, we construct the rooted derived graph $\g_{(v)}$ by rooting $\g_{\b_i}'$ at $v$ (and keeping the unlabelled vertex).
    \item For $v$ a vertex in $\varphi$, in increasing order of label in $\varphi$, let $\t_v := \TT_{\g_{(v)}}$ be the decorated block tree on this rooted subgraph with an unlabelled vertex. Add $\t_v$ as the rightmost child of $r_{\TT}$.
\end{enumerate}

\subsection{Motivation and background}

Planar maps and planar graphs have received much attention from enumerative and probabilistic points of view. The study of planar map enumeration started with the groundbreaking combinatorial work of Tutte, who gave a close formula for the number of rooted planar maps of size $n$ in the 1960s \cite{tutte_1963}. Tutte's work prompted many perspectives and extensions since \cite{ChassaingSchaeffer2004,cori_vauquelin_1981,LeGall2013,Miermont2013,Schaeffer1998}. Despite extensive studies, the asymptotic number of planar graphs were successfully determined only much more recently by Giménez and Noy \cite{GN09} in 2009, building on work of Bender, Gao, and Wormald \cite{bender2002number}.

Many results on planar maps and planar graphs rely on applying decompositions that work in tree-like fashion, allowing combinatorial properties to be deduced. A prominent example of this theme is the \emph{block decomposition}. For both planar maps and connected planar graphs, there is a unique way to decompose them into maximal $2$-connected submaps or subgraphs, called \emph{blocks}  (see, e.g., \cite{diestel2025graph, harary1969,tutte_1963}). In both cases, one can construct tree-like structures by viewing the blocks as vertices of a graph whose edges join overlapping blocks.

In the case of planar maps, such a tree was described by Addario-Berry \cite{2Louigi}, following Tutte's description of the block decomposition \cite{tutte_1963}. Moreover, Addario-Berry \cite{2Louigi} showed that this tree could be used to obtain information about \emph{random} planar maps: Since the block tree of a uniform planar map of size $n$ follows the law of a \emph{\BGW{} tree}\footnote{Also called \emph{Galton--Watson tree} or \emph{Bienaymé--Galton--Watson tree}, we adopt the change of notation proposed by Addario-Berry, Brandenberger, Hamdan, and Kerriou \cite{ABBHK22}.}, whose properties are well-known, properties on the sizes of the $2$-connected components of a random map follow. It was subsequently shown that this approach fruitfully allows one to consider \emph{block-weighted models}: instead of sampling according to a uniform law, one considers a parameter $u>0$, and samples a map of size $n$ with probability proportional to $u$ to the power of its number of blocks. (Note that $u=1$ corresponds to the usual uniform law.) Fleurat and the second author \cite{FleuratSalvy23} showed that the block tree of such a random planar map still follows a \BGW{} law, and used this to obtain results on the sizes of blocks of such maps as well as scaling limits. More precisely, they discovered a phase transition between the universality classes of maps (converging to the Brownian sphere) and plane trees (converging to the Brownian tree), depending on the value of $u$, obtaining precise results on block sizes in each phase.

Using an extensive list of decomposition of maps into subfamilies \cite{airy}, the second author \cite{Salvy23,Sal24} showed that this approach can be successfully adapted to deal with different decompositions of maps, giving a very similar phase transition between a ``map phase'' and a ``tree phase''. Albenque, Fusy, and the second author \cite{AFS24} used this approach for a family of \emph{decorated} maps, which are planar maps endowed with one of their spanning tree. They found a phase transition between a ``tree-rooted map phase'' and a ``tree phase''.

Since this technique of using block decompositions to obtain interesting results for planar \emph{maps} is robust, it is intriguing to extend this approach to connected planar \emph{graphs}. The planar graph case is slightly more intricate than the planar map case because one has to take into account the labelling and cannot rely on an embedding as in the map case.

\subsection{Main results}
The main contributions of this paper are two-fold. First, we show that a random connected rooted labelled  graph from a block-stable class (e.g., a random connected planar graph), which is sampled according to a block-weighted law, exhibits a phase transition (see \cref{thm:phase-transition}). To this end, we introduce a \emph{decorated block tree} that encodes a block decomposition of such a connected graph (see \cref{sec:decorated}). This extends the lines of previous study based on a block tree decomposition from planar \emph{maps} to connected planar \emph{graphs}. Because block-stable classes are very general, our results are also very general, but not necessarily precise.

Second, we show how the block tree methodology can be used to obtain more precise results. First, we use it to obtain enumeration results for the block-weighted generating function. Then we determine the distribution of block sizes of random block-weighted planar graphs and block-stable graphs of the same singular type (see \cref{thm:block-sizes}). As a result, we obtain more precise information on block size of a uniform random connected planar graph than what was previously known in the literature \cite{GNR13, PS11}.

\subsubsection{Phase transition in block-weighted block-stable graphs}
A class $\cC$ of connected graphs is said to be \emph{block-stable} if for each graph $\g \in \cC$ each of its $2$-connected components (called \emph{blocks}) belongs to $\cC$. Write $\cB$ for the class of $2$-connected elements of $\cC$. Write $B(y)$ and $C(x)$ for the corresponding exponential generating functions (EGFs), and set $C^\bullet(x) = x C(x)$. It was shown, e.g., by Robinson \cite[Theorem 4]{Rob70}, that
\begin{equation}
\label{eq:Cbullet-block}
C^\bullet(x) = x \exp(B'(C^\bullet(x))).
\end{equation}
As we will see in \cref{subsec:anal-comb-block-tree}, this equation can be translated into a ``decorated'' block tree.

More precisely, let $C^\bullet(x,u)$ be the exponential generating function for \emph{connected rooted} graphs of $\cC$, where $x$ marks vertices and $u$ marks blocks. Write $\rho(u)$ for the radius of convergence of $C^\bullet(\cdot,u)$. For $u>0$, we consider the following \emph{block-weighted} probability laws on $\cC^\bullet$: for a graph $\g \in \cC$ of size $|\g|$ with $b(\g)$ many $2$-connected components, we set
\[\pr[_{u}]{\g} := \frac{\rho(u)^{|\g|}u^{b(\g)}}{|\g|!C^\bullet(\rho(u),u)},\qquad\text{and}\qquad \pr[_{n,u}]{\g} := \frac{u^{b(\g)}}{[x^n] C^\bullet(x,u)}\ind_{|\g|=n}.\]

We show that, when considering a block-weighted law on a connected rooted graphs from a block-stable classes (including connected rooted planar graphs), this block tree follows a \BGW{} tree distribution (see \cref{prop:law-TG}). Our first main theorem is about phase transitions in the criticality of this \BGW{} tree.

In the case of planar maps, such a tree was described by Addario-Berry \cite{2Louigi}, following Tutte's description of the block decomposition \cite{tutte_1963}. Moreover, Addario-Berry \cite{2Louigi} showed that this tree could be used to obtain information about \emph{random} planar maps: Since the block tree of a uniform planar map of size $n$ follows the law of a \emph{\BGW{} tree}\footnote{Also called \emph{Galton--Watson tree} or \emph{Bienaymé--Galton--Watson tree}, we adopt the change of notation proposed by Addario-Berry, Brandenberger, Hamdan, and Kerriou \cite{ABBHK22}.}, whose properties are well-known, properties on the sizes of the $2$-connected components of a random map follow. It was subsequently shown that this approach fruitfully allows one to consider \emph{block-weighted models}: instead of sampling according to a uniform law, one considers a parameter $u>0$, and samples a map of size $n$ with probability proportional to $u$ to the power of its number of blocks. (Note that $u=1$ corresponds to the usual uniform law.) Fleurat and the second author \cite{FleuratSalvy23} showed that the block tree of such a random planar map still follows a \BGW{} law, and used this to obtain results on the sizes of blocks of such maps as well as scaling limits. More precisely, they discovered a phase transition between the universality classes of maps (converging to the Brownian sphere) and plane trees (converging to the Brownian tree), depending on the value of $u$, obtaining precise results on block sizes in each phase.

\subsection{Main results}
The main contributions of this paper are two-fold. First, we show that a random connected rooted labelled graph from a block-stable class (e.g., a random connected planar graph), which is sampled according to a block-weighted law, exhibits a phase transition (see \cref{thm:phase-transition}). To this end, we introduce a \emph{decorated block tree} that encodes a block decomposition of such a connected graph (see \cref{sec:decorated}). This extends the lines of previous study based on a block tree decomposition from planar \emph{maps} to connected planar \emph{graphs}. Because block-stable classes are very general, our results are also very general, but not necessarily precise.

Second, we show how the block tree methodology can be used to obtain more precise results. First, we use it to obtain enumeration results for the block-weighted generating function. Then we determine the distribution of block sizes of random block-weighted planar graphs and block-stable graphs of the same singular type (see \cref{thm:block-sizes}). As a result, we obtain more precise information on block size of a uniform random connected planar graph than what was previously known in the literature \cite{GNR13, PS11}.

\subsubsection{Phase transition in block-weighted block-stable graphs}
A class $\cC$ of connected graphs is said to be \emph{block-stable} if for each graph $\g \in \cC$ each of its $2$-connected components (called \emph{blocks}) belongs to $\cC$. Write $\cB$ for the class of $2$-connected elements of $\cC$. Write $B(y)$ and $C(x)$ for the corresponding exponential generating functions (EGFs), and set $C^\bullet(x) = x C(x)$. It was shown, e.g., by Robinson \cite[Theorem 4]{Rob70}, that
\begin{equation}
\label{eq:Cbullet-block}
C^\bullet(x) = x \exp(B'(C^\bullet(x))).
\end{equation}
As we will see in \cref{subsec:anal-comb-block-tree}, this equation can be translated into a ``decorated'' block tree.

More precisely, let $C^\bullet(x,u)$ be the exponential generating function for \emph{connected rooted} graphs of $\cC$, where $x$ marks vertices and $u$ marks blocks. Write $\rho(u)$ for the radius of convergence of $C^\bullet(\cdot,u)$. For $u>0$, we consider the following \emph{block-weighted} probability laws on $\cC^\bullet$: for a graph $\g \in \cC$ of size $|\g|$ with $b(\g)$ many $2$-connected components, we set
\[\pr[_{u}]{\g} := \frac{\rho(u)^{|\g|}u^{b(\g)}}{|\g|!C^\bullet(\rho(u),u)},\qquad\text{and}\qquad \pr[_{n,u}]{\g} := \frac{u^{b(\g)}}{[x^n] C^\bullet(x,u)}\ind_{|\g|=n}.\]

We show that, when considering a block-weighted law on a connected rooted graphs from a block-stable classes (including connected rooted planar graphs), this block tree follows a \BGW{} tree distribution (see \cref{prop:law-TG}). Our first main theorem is about phase transitions in the criticality of this \BGW{} tree.

\subsubsection{Phase transition in block-weighted block-stable graphs}
A class $\cC$ of connected graphs is said to be \emph{block-stable} if for each graph $\g \in \cC$ each of its $2$-connected components (called \emph{blocks}) belongs to $\cC$. Write $\cB$ for the class of $2$-connected elements of $\cC$. Write $B(y)$ and $C(x)$ for the corresponding exponential generating functions (EGFs), and set $C^\bullet(x) = x C(x)$. It was shown, e.g., by Robinson \cite[Theorem 4]{Rob70}, that
\begin{equation}
\label{eq:Cbullet-block}
C^\bullet(x) = x \exp(B'(C^\bullet(x))).
\end{equation}
As we will see in \cref{subsec:anal-comb-block-tree}, this equation can be translated into a ``decorated'' block tree.

We show that, when considering a block-weighted law on a connected rooted graphs from a block-stable classes (including connected rooted planar graphs), this block tree follows a \BGW{} tree distribution (see \cref{prop:law-TG}). Our first main theorem is about phase transitions in the criticality of this \BGW{} tree.

For a more precise version of this theorem, see \cref{th:phase-trans-singular,thm:phasetransition_block_weighted}. For the special case of planar graphs (and similar classes), we also show that the subcritical case corresponds to a ``rooted graph phase'' while the supercritical case corresponds to a ``tree phase''. Our key proof techniques build upon the methods previously developed by Fleurat and the second author for block-weighted planar maps \cite{FleuratSalvy23}, where analogous behaviour was exhibited.

\section{Block-weighted laws for block-stable classes}
\label{sec:block-weighted}
In this section, we study the main objects of this paper, namely \emph{block-weighted} connected graphs. We recall that these are graphs sampled according to a block-weighted law. As explained in the introduction, let $C^\bullet(x,u)$ be the exponential generating function for \emph{connected rooted} graphs of $\cC$, where $x$ marks vertices and $u$ marks blocks. Let $\rho(u)$ denote the radius of convergence of $C^\bullet(\cdot,u)$. From now on, we always assume $u>0$. We consider the following \emph{block-weighted} probability laws on $\cC^\bullet$: For a graph $\g \in \cC^\bullet$ with $b(\g)$ many $2$-connected components, we set
\[\pr[_{u}]{\g} := \frac{\rho(u)^{|\g|}u^{b(\g)}}{|\g|!C^\bullet(\rho(u),u)}\qquad\text{and}\qquad \pr[_{n,u}]{\g} := \frac{u^{b(\g)}}{[x^n] C^\bullet(x,u)}\ind_{|\g|=n}.\]

\begin{proposition}
\label{prop:law-TG}
Fix $u>0$, let $\G$ be distributed according to $\Pu$, and denote by $(\bTT,(\bphi_v,v\in \bTT))$ the decorated block tree $\TT_\G$, where $\bTT$ is the undecorated block tree and $(\bphi_v,v\in \bTT)$ is its family of vertex decorations. Then the law of $\TT_\G$ can be described as follows:
\begin{itemize}
\item $\bTT$ follows the law $\BGWl{\muT^u}$;
\item Conditionally given $\bTT=\t$, the decorations $(\bphi_{v},v\in \t)$ are independent random variables, and, for $v\in\t$, $\bphi_{v}$ is distributed according to $\Gamma \Phi(C^\bullet(\rho(u),u),u)$ conditioned to have size $d_\t(v)$.
\end{itemize}
For every $n\in\Npos$, the same statements hold under $\Pnu$, only replacing $\BGWl{\muT^u}$ with $\BGWl{\muT^u,n}$.
\end{proposition}

\begin{theorem}\label{th:phase-trans-singular}
Let $\cC$ be a block-stable class of connected graphs, let $\cB$ be its class of $2$-connected components with EGF $B(y)$, and let $\rho_B$ denote the radius of convergence of $B$. Let
\[u_C := \frac{1}{\rho_B B''(\rho_B)}.\]
Then the radius of convergence $\rho(u)$ of the series $C^\bullet(\cdot,u)$ (on block-weighted rooted elements of $\cC$) and $y(u) = C^\bullet(\rho(u),u)$ satisfy the following:
\begin{itemize}
 \item If $u\leq u_C$, then
 \[\rho(u) = \frac{\rho_B}{\exp(uB'(\rho_B))}\qquad\text{and}\qquad y(u)=\rho_B;\]
 \item If $u\geq u_C$, then
 \[\rho(u) = \frac{1}{uB''(y(u))\exp(uB'(y(u)))}\qquad\text{and}\qquad y(u)=\frac{1}{uB''(y(u))}.\]
\end{itemize}
\end{theorem}

\begin{theorem}\label{thm:phasetransition_block_weighted}
Let $\Gnu$ be a graph of size $n$ from a block stable class distributed according to $\Pnu$. Then its block tree $\Tnu$ follows the law $\BGWl{\muT^u, n}$ where $\muT^u$ satisfies the following.
\begin{itemize}
 \item If $u<u_C$, then
 \begin{equation}
 \label{eq:exp-muu}
 \E{\muT^u} = \frac{u}{u_C}<1,
 \end{equation} so the \BGW{} law is subcritical;
 \item If $u\geq u_C$, then \[\E{\muT^u} = 1,\] so the \BGW{} law is critical.
\end{itemize}
\end{theorem}

We want to understand the behaviour of an element of the class $\Phi$ with a fixed size. We use the fact that the function $\Phi_u := \Phi(\cdot, u)$ can be written as
\[\Phi_u(y) = V(W(y)),\]
for $V (\cdot) := \exp (\cdot) $ and $W (\cdot) := uB'(\cdot)$. The \emph{Gibbs partition} framework allows one to study the size of the elements of the class $\cW$ (whose EGF is given by $W (\cdot)$) in a random element $\varphi$ drawn from a Boltzmann law on $\Phi_u$. In particular, \cite[Theorem 3.11]{Stu24} allows us to prove the following proposition.
\begin{proposition}
\label{prop:gibbs-partition}
Let $\mnu{j}$ denote the size of a largest element in the set decorating $\vnu{j}$ in $\Tnu$. Let $R$ be a random variable with probability generating function
\[P_R(z) := \E{z^R} = \frac{\exp(uB'(\rho_Bz))}{\exp(uB'(\rho_B))}.\]
Then for any $j \in \mathbb{N}_0$, it holds that
\[\bdTnu{j} - \mnu{j} \xrightarrow[n\to\infty]{(d)} R,\]
so, in particular,
\[\bdTnu{j} - \mnu{j} = O_{\mathbb{P}}(1)\qquad\text{as}\quad n\to\infty.\]
\end{proposition}
\begin{proof}
The collection of block sizes in a random decoration of a vertex of outdegree $k$ in $\Tnu$ can be interpreted as a Gibbs partition. As described in \cite[Theorem 3.11]{Stu24}, under certain conditions a Gibbs partition of size $k$ has a unique giant component whose size $\bm_k$ satisfies
\begin{equation}
\label{eq:gibbs-result}
k - \bm_k \xrightarrow[k\to\infty]{(d)} S \coloneqq \sum_{i=1}^{\hat{N} - 1} X_i.
\end{equation}
Here, the $X_i$'s are independent copies of a random variable $X$ whose probability generating function is
\[P_X(z) \coloneqq \mathbb{E}[z^X] = \frac{W(\rho_W z)}{W(\rho_W)},\]
and $\hat{N}$ is a random variable whose probability generation function is
\[P_{\hat{N}}(z) \coloneqq \mathbb{E}[z^{\hat{N}}] = \frac{V'(W(\rho_W)z)z}{V'(W(\rho_W))}.\]
One can check that the probability generating function of the random variable $S$ is
\[P_S(z) \coloneqq \mathbb{E}[z^S] = \frac{P_{\hat{N}}(P_X(z))}{P_X(z)} = \frac{\exp(uB'(\rho_Bz))}{\exp(uB'(\rho_B))},\]
so $S$ and $R$ have the same distribution. \Cref{prop:gibbs-partition} is a then consequence of:
\begin{itemize}
 \item Stufler \cite[Theorem 3.11]{Stu24};
 \item For all $j\in\Npos$, $\bdTnu{j}$ converges in probability to $+\infty$;
 \item The fact that one can sample $\Gnu$ according to $\Pnu$ by first sampling $\Tnu$, then its decorations (see \cref{prop:law-TG}).
\end{itemize}

For a more precise version of this theorem, see  \cref{th:phase-trans-singular,thm:phasetransition_block_weighted}. For the special case of planar graphs (and similar classes), we also show that the subcritical case corresponds to a ``rooted graph phase'' while the supercritical case corresponds to a ``tree phase''. Our key proof techniques build upon the methods previously developed by Fleurat and the second author for block-weighted planar maps \cite{FleuratSalvy23}, where analogous behaviour was exhibited.

It is well-known that the relative positions of $\rho_B B''(\rho_B)$ and $1$ allow us to distinguish (sub)criticality (see, e.g., \cite{DFKKR11}). More precisely, when $\rho_B B''(\rho_B) > 1$, it is known that the graphs behave ``tree-like'', which is consistent with $u_C < 1$ and the uniform case being supercritical. On the contrary, when $\rho_B B''(\rho_B) < 1$, the graphs behave as general planar graphs do, which is consistent with $u_C > 1$.

We note that block-weighted block-stable graph classes were already considered by Stufler \cite[\S5]{Stu18b} in a general setting using the framework of Gibbs partitions, but our current work has a different focus.

\subsubsection{Enumeration and block sizes in block-weighted planar graphs}
Our next results focus on the case where $B'$ has a singularity of type $3/2$ (which encompasses planar graphs). Under this assumption, we also show how the phase transition described in \cref{thm:phase-transition} affects the behaviour of $[x^n] C^\bullet(x,u)$, as well as the typical sizes of blocks. The following is our main result on the former.

\begin{theorem}\label{thm:enum} For every $u>0$, there exist $c > 0$, $c_1(u)>0$, and $c_2(u)>0$ such that:
\begin{itemize}
    \item in the subcritical phase $u < u_C$,
    \[[x^n] C^\bullet(x,u) \simn c_1(u) n^{-5/2} \rho(u)^{-n};\]
    \item in the critical phase $u = u_C$,
    \[[x^n] C^\bullet(x,u) \simn c n^{-5/3} \rho(u_C)^{-n};\]
    \item in the supercritical phase $u > u_C$,
    \[[x^n] C^\bullet(x,u) \simn c_2(u) n^{-3/2} \rho(u)^{-n}.\]
\end{itemize}
\end{theorem}

For a more precise version of this theorem, see \cref{th:dvp-sing-Cbxu}. Thus, in the subcritical phase, the polynomial correction is that of rooted graphs: $n^{-5/2}$. Conversely, in the supercritical phase, the correction is that of plane trees: $n^{-3/2}$. This justifies why one can view the subcritical phase as a ``general rooted graph phase'' and the supercritical phase as a ``tree phase''.  It is worth noting that in the critical phase, i.e., when $u = u_C$, a novel asymptotic behaviour emerges, characterised by a polynomial correction of $n^{-5/3}$.

The next theorem summarises our main results on the typical block sizes in the three different phases.

\begin{theorem}\label{thm:block-sizes}
There is a phase transition for block sizes depending on the value of $u$:
\begin{itemize}
    \item In the subcritical phase $u < u_C$, there is a largest block of size $\Theta_\mathbb{P}(n)$ and a second largest block of size $\Theta_\mathbb{P}(n^{2/3})$;
    \item In the critical phase $u = u_C$, there is a largest block of size $\Theta_\mathbb{P}(n^{2/3})$ and likewise for the next largest block sizes;
    \item In the supercritical phase $u > u_C$, there is a largest block of size $\Theta_\mathbb{P}(\ln (n))$ and likewise for the next largest block sizes.
\end{itemize}
\end{theorem}

As we will see in \cref{thm:phasetransition_block_sizes}, we also determine precise limit laws for the block sizes when appropriately scaled, in the subcritical and critical cases,  and provide a more precise description of what happens in the supercritical case. Our results extend some previous work in the literature \cite{GNR13,PS11} from uniform to block-weighted random planar graphs. In fact, the above two results hold for any block-stable class where $B'$ is \emph{singular with exponent $3/2$} (for a definition, see e.g., \cite[Equation (46)]{airy}). This includes the class of planar graphs as shown by Bender, Gao, and Wormald \cite{bender2002number} (see \cref{eq:asymp-exp-B'} for the singular expansion of $B'$).

\subsection{Key proof techniques and related work}

The primary proof methods used in this paper build on those used for block-weighted planar maps in \cite{FleuratSalvy23}, with some ideas originating from \cite{2Louigi}. The Lagrangean form of \cref{eq:Cbullet-block} implicitly gives a tree-like structure and we make it combinatorially explicit in the form of a \emph{decorated block tree}, detailed in \cref{subsec:anal-comb-block-tree}. Our decorated block tree looks similar to the usual notion of a block decomposition of a connected graph, but is different in that the vertices of our tree represent cut-vertices instead of blocks, and these cut-vertices are decorated by sets of blocks. We use tools from singularity analysis and results about \BGW{} trees to pinpoint and describe a phase transition, as in \cref{thm:phase-transition,thm:enum}.

To prove our results about typical block sizes in random block-weighted planar graphs, as in \cref{thm:block-sizes}, we also incorporate some ideas on planar maps from \cite{FleuratSalvy23}, but extra steps are needed due to the fact that our block tree decorations are not blocks, but \emph{sets of blocks}, or more precisely, elements of the class $\Set\circ \mathcal{B}'$. First we apply known results on degree distributions of \BGW{} trees (by, e.g., Janson \cite{survey-trees}) to our block tree. Then we combine this with results on Gibbs partitions by Stufler \cite{Stu24} to deduce typical block sizes. Similar ideas were used to study block-weighted simple triangulations in \cite[\S4.2]{Sal24}, where the decorations were sequences of blocks.

Finally, we mention some other related works. In the uniform case, Panagiotou and Steger \cite{PS11} proved a dichotomy theorem on the behaviour of the size of a largest block in connected graph classes: For classes similar to planar graphs there is a unique largest block of linear size almost surely, whereas for classes similar to series-parallel graphs all components are at most logarithmic in size almost surely. Gim\'enez, Noy, and Ru\'e \cite[Theorem 5.4]{GNR13} subsequently refined the dichotomy theorem of Panagiotou and Steger \cite{PS11} and also showed a limiting distribution on the sizes of the largest blocks, involving the Airy distribution. As mentioned earlier, our work considers in particular planar graphs with block weights and thus generalises some of the results in the works above.

\subsection{Plan of the paper}
In \cref{sec:preliminaries}, we give notation and recall usual notions of block decompositions of graphs. In \cref{sec:decorated}, we detail the construction of the decorated block tree of a rooted graph. In \cref{sec:block-weighted}, we introduce the block-weighted model for block-stable graph classes and obtain results about it. In \cref{sec:block-sizes}, we focus on classes where $B'$ is singular with exponent $3/2$ (which includes planar graphs), and obtain results on enumeration and block sizes. In \cref{sec:conclusion}, we discuss possible extensions and future work.

The post-theorem text does not give a step-by-step proof of Theorem~\ref{thm:phase-transition}, but it does outline the main ingredients used to establish it. The argument is said to build on methods for block-weighted planar maps \cite{FleuratSalvy23}. The key structural input is that “The Lagrangean form of \cref{eq:Cbullet-block} implicitly gives a tree-like structure” which the authors “make … combinatorially explicit in the form of a \emph{decorated block tree}.” This tree is “different in that the vertices of our tree represent cut-vertices instead of blocks, and these cut-vertices are decorated by sets of blocks.” With this explicit tree representation in hand, they “use tools from singularity analysis and results about \BGW{} trees to pinpoint and describe a phase transition, as in \cref{thm:phase-transition,thm:enum},” yielding the subcritical/critical dichotomy for the block tree law.

They also connect the phase transition threshold to a known (sub)criticality criterion: “It is well-known that the relative positions of $\rho_B B''(\rho_B)$ and $1$ allow us to distinguish (sub)criticality,” with the interpretation that when $\rho_B B''(\rho_B) > 1$ the graphs behave “tree-like” (consistent with $u_C<1$), while when $\rho_B B''(\rho_B) < 1$ they behave like “general planar graphs” (consistent with $u_C>1$).

The post-theorem text does not give a step-by-step proof of the main theorem, but it does outline the main ingredients used to establish it. The argument is said to build on methods for block-weighted planar maps Fleurat and Salvy, title unknown (2023). The key structural input is that “The Lagrangean form of
\begin{equation}
\label{eq:Cbullet-block}
C^\bullet(x) = x \exp(B'(C^\bullet(x))).
\end{equation}
implicitly gives a tree-like structure” which the authors “make … combinatorially explicit in the form of a \emph{decorated block tree}.” This tree is “different in that the vertices of our tree represent cut-vertices instead of blocks, and these cut-vertices are decorated by sets of blocks.” With this explicit tree representation in hand, they “use tools from singularity analysis and results about \BGW{} trees to pinpoint and describe a phase transition,” yielding the subcritical/critical dichotomy for the block tree law.

They also connect the phase transition threshold to a known (sub)criticality criterion: “It is well-known that the relative positions of $\rho_B B''(\rho_B)$ and $1$ allow us to distinguish (sub)criticality,” with the interpretation that when $\rho_B B''(\rho_B) > 1$ the graphs behave “tree-like” (consistent with $u_C<1$), while when $\rho_B B''(\rho_B) < 1$ they behave like “general planar graphs” (consistent with $u_C>1$).

Question:
Let \(\mathcal C\) be a block-stable class of connected labelled graphs, meaning that every 2-connected component (block) of every graph in \(\mathcal C\) again belongs to \(\mathcal C\). Let \(\mathcal B\) be the class of 2-connected graphs in \(\mathcal C\), let \(B(y)\) be its exponential generating function, and let \(\rho_B\) be the radius of convergence of \(B\). Define
\[
u_C:=\frac{1}{\rho_B B''(\rho_B)}.\]
For \(u>0\), let \(C^{\bullet}(x,u)\) be the exponential generating function of connected rooted graphs in \(\mathcal C\), where \(x\) marks vertices and \(u\) marks blocks. For each \(n\), let \(\mathbb P_{n,u}\) be the block-weighted law on rooted connected graphs of size \(n\), assigning probability proportional to \(u^{b(G)}\), where \(b(G)\) is the number of blocks of \(G\). The associated block tree is the tree encoding the block decomposition of such a sampled graph. Which statement holds for every such class \(\mathcal C\)?

The singular behaviour of \(C^{\bullet}(x,u)\) depends on whether \(u<\nu_C\) or \(u\ge \nu_C\); moreover, if a graph is sampled according to \(\mathbb P_{n,u}\), then its block tree has a subcritical Bienaymé–Galton–Watson law when \(u<\nu_C\), and a critical Bienaymé–Galton–Watson law when \(u\ge \nu_C\).

Choices:
B. The singular behaviour of \(C^{\bullet}(x,u)\) depends on whether \(u<\nu_C\) or \(u>\nu_C\); moreover, if a graph is sampled according to \(\mathbb P_{n,u}\), then its block tree has a subcritical Bienaymé–Galton–Watson law when \(u\le \nu_C\), and a critical Bienaymé–Galton–Watson law when \(u>\nu_C\).
C. If a graph is sampled according to \(\mathbb P_{n,u}\), then its block tree has a subcritical Bienaymé–Galton–Watson law when \(u<\nu_C\), and a critical Bienaymé–Galton–Watson law when \(u\ge \nu_C\).
D. The singular behaviour of \(C^{\bullet}(x,u)\) depends on whether \(u<\nu_C\) or \(u\ge \nu_C\); moreover, if a graph is sampled according to \(\mathbb P_{n,u}\), then its block tree has a subcritical Bienaymé–Galton–Watson law whenever \(u<\nu_C\), and there exists a critical Bienaymé–Galton–Watson law at \(u=\nu_C\) only, while for every \(u>\nu_C\) the law is no longer critical.
E. The singular behaviour of \(C^{\bullet}(x,u)\) depends on whether \(u<\nu_C\) or \(u\ge \nu_C\); moreover, if a graph is sampled according to \(\mathbb P_{n,u}\), then its block tree has a critical Bienaymé–Galton–Watson law when \(u<\nu_C\), and a subcritical Bienaymé–Galton–Watson law when \(u\ge \nu_C\).

boundary case at the threshold \(u=\nu_C\)

drops the singular-behaviour clause while retaining the BGW phase-transition conclusion

uniform criticality for all \(u\ge \nu_C\)

subcritical/critical correspondence relative to the threshold

The stem explicitly defines the quantity 1/(rho_B B''(rho_B)) under a different name (nu_C), and the correct option uses exactly that expression as the threshold. This does not fully reveal the whole answer, but it gives a strong hint about the critical value.

The item is close to a theorem-recall question: it asks for the stated phase classification associated with a previously defined threshold. However, it is not a pure tautology because the options differ on boundary inclusion, critical vs supercritical behaviour, and whether the BGW clause is complete.

Some reasoning is needed to distinguish subtle alternatives, especially the treatment of u = u_C and the critical/subcritical classification. Still, the question mainly tests precise recall of a known result rather than generating a conclusion from first principles.

The distractors are plausible and mathematically targeted: one flips the boundary case, one omits an essential clause, one substitutes supercritical for critical, and one uses the reciprocal threshold incorrectly. These reflect realistic failure modes.

A decent but theorem-close MCQ: distractors are strong, but the stem leaks the critical expression and the task is largely precise recall rather than genuinely generative reasoning.


\end{center}

\mdseries
A central problem in algebraic geometry is to determine the extent to which a given algebraic variety $X$ admits rationality properties—such as rationality, unirationality, or rational connectedness. Within the past few years, natural invariants which measure these rationality properties have been extensively studied. The foundational case is the one-dimensional setting. Consider $C$, an irreducible smooth curve. The gonality of $C$ is defined as the minimal degree of a morphism $\phi : C \rightarrow \mathbb{P}^{1}$ and we denote it $\gonality(C)$. In the higher dimensional setting, various extensions of this definition exist, depending on the rationality property of interest. Consider a smooth projective variety $X$ of dimension $n$. On the one hand, the degree of irrationality of $X$ is defined as the minimal degree of a rational dominant map $f : X \dashrightarrow \mathbb{P}^{n}$. We denote it $\irr(X)$. Then, $X$ is rational if and only if $\irr(X)=1$. This definition coincides with the gonality in the one-dimensional case. On the other hand, the covering gonality of $X$ is defined as the minimal gonality of a curve $C$ passing through a general point $x \in X $. We denote it $\covgon(X)$. One can notice that $X$ is uniruled if and only if $\covgon(X)=1$. \\
Historically, the first result is due to Max Noether who showed that the gonality of a smooth plane curve $C$ of degree $d\geq 3$ is computed by the projection from a point $x\in C$. This result was subsequently generalized in \cite{degreehypersurfaces} for a very general irreducible smooth hypersurface $X\subset\mathbb{P}^{n}$ of degree $d\geq 2n+1$. The main result of \cite{degreehypersurfaces} is that $\irr(X)$ is computed by the projection from a point $x\in X$. More generally, these invariants are actively studied in different settings, see e.g., \cite{chen2020fano}, \cite{stapleton2020degree}, \cite{chen2023rational}, \cite{chen2021multiplicative}, \cite{colombo2022degree}, \cite{bastianelli2022covering}, \cite{moretti2024degree}, \cite{voisin2022fibrations} and for a global survey the reader can consult \cite{chen2025primermeasuresirrationality}. In this note, we are interested in the covering gonality. In recent years, several results have been established in various contexts. For instance, in \cite{martin2020conjecture}, Martin exhibits a lower bound for the covering gonality of a very general abelian variety in any dimension answering a conjecture of Voisin in \cite{voisin2018chow}. Also, in \cite{chen2021multiplicative}, the author gives a multiplicative lower bound for the covering gonality of certain complete intersections. Finally, in \cite{bastianelli2019gonality} the authors compute the covering gonality of a very general irreducible smooth hypersurface $X\subset \mathbb{P}^{n+1}$ of degree $d\geq 2n+2$. This work actually originated in \cite{lopez1995curves}, \cite{degreehypersurfaces} in which the authors investigated the lower dimensional cases.

Inspired by the result \cite[Theorem 1.1]{bastianelli2019gonality}, we prove the following theorem:

\vspace{0.35cm}

\vspace{0.35cm}

\begin{thm}
Let $X, Y \subset \mathbb{P}^{n+1} \times \mathbb{P}^{m+1}$ be very general smooth hypersurfaces
of bi-degrees $(a_1,b_1)$ and $(a_2,b_2)$ respectively with $a_{1},a_{2} \geq 2n+m+2$ and $b_{1},b_{2} \geq 2m+n+2$. Then:
\vspace{0.5cm}
\begin{center}
 $\operatorname{cov.gon}(X,Y) \;\geq\; k_1 + k_2 - n - m$
\end{center}
where :
\[
k_1 = \min\{a_1 - n - 3, b_1 - m - 3\}, \quad
k_2 = \min\{a_2 - n - 3, b_2 - m - 3\}.
\]

\begin{prop}
\label{Estimate}
Let $a\geq 2n+m+1,b\geq 2m+n+1$ and $B$ the congruence of lines defined of the covering family of curves of gonality $c$ as defined above. First, either $\mathcal{B} \subset Gr(1,n+1) \times \mathbb{P}^{m+1}$ or $B\subset \mathbb{P}^{n+1} \times Gr(1,m+1)$ and if $B \subset Gr(1,n+1) \times \mathbb{P}^{m+1}$ (resp. $\mathcal{B}\subset \mathbb{P}^{n+1} \times Gr(1,m+1)$), then :
\begin{center}
 $c\geq a-n$ (resp, $c\geq b-m$).
\end{center}
\end{prop}
\begin{proof}
 First, as $\mathcal{B}$ is an irreducible subvariety of $ Gr(1,n+1)\times \mathbb{P}^{m+1} \sqcup \mathbb{P}^{n+1}\times Gr(1,m+1)$, the irreducibility of $\mathcal{B}$ implies that either $\mathcal{B} \subset Gr(1,n+1)\times \mathbb{P}^{m+1}$ or $\mathcal{B} \subset \mathbb{P}^{n+1}\times Gr(1,m+1)$. Suppose now that $\mathcal{B} \subset Gr(1,n+1)\times \mathbb{P}^{m+1}$ and that $c\leq a-n-1$. Then, for $t \in T, y\in \mathbb{P}^{1}$, the fibre $\phi^{-1}(t,y)$ lie on a line $l_{(t,y)}$ of the form $(l'_{(t,y)},z) \subset \mathbb{P}^{n+1} \times \mathbb{P}^{m+1}$, where $z\in \mathbb{P}^{m+1}$ and $l'_{(t,y)} \subset \mathbb{P}^{n+1}$ is a line. Moreover, it must satisfy the Cayley-Bacharach condition with respect to $\vert K_{X} \vert$. But $X_{z}$ is a hypersurface of degree $a$ in $\mathbb{P}^{n+1}$, hence $K_{X_{z}} = (a-n-2)H$ where $H$ is the hyperplane divisor of $\mathbb{P}^{n+1}$. Then, one can produce $E' \in \vert K_{X_{z}} \vert$ such that $E'$ passes through all the points in $pr_{1}(\phi^{-1}(t,y))$ but one as there are at most $a-n-1$ points in $pr_{1}(\phi^{-1}(t,y))$. Then, we can produce a divisor $E \in \vert K_{X} \vert \cong \vert \mathcal{O}(a-n-2,b-m-2)\vert $ such that $E$ passes through any point of $\phi^{-1}(t,y)$ but one. Indeed, consider a divisor $E'' \in \vert \mathcal{O}_{\mathbb{P}^{n+1}\times \mathbb{P}^{m+1}}(b-m-2) \vert $ passing through $w\in \mathbb{P}^{m+1}$. Then, the divisor $E'\boxtimes E''$ belongs to $\vert K_{X} \vert $ and passes through every points of $\phi^{-1}(t,y)$ but one. This implies that the fibre does not respect the Cayley-Bacharach condition with respect to $\vert K_{X} \vert$, which is a contradiction. Then $c\geq a-n$. The proof can be done the other way around to obtain that if $\mathcal{B} \subset \mathbb{P}^{n+1} \times Gr(1,m+1)$, then $c \geq b-m$.
\end{proof}

\begin{prop}
\label{LocalisationOfCurves}
Let $X$ be a very general and smooth hypersurface of bi-degree $(a,b)$ in $\mathbb{P}^{n+1} \times \mathbb{P}^{m+1}$, such that $a \geq 2n+m+1$ and $b\geq 2m+n+1$. Consider a covering family of c-gonal curve over $X$ with $c \leq k-3$. Then for each $t \in T$ there exists $x_{t} \in f(\mathcal{C}_{t})$ such that $f(\mathcal{C}_{t}) \subset V_{x_{t}}^{a-c}:=\{(\ell,z) \in Gr(1,n+1)\times\mathbb{P}^{m+1}, (z,\ell) \cdot X \geq (a-c)x_{t}\}$ or $f(C_{t}) \subset {W}^{b-c}_{x_{t}} : = \{\ell \in \mathbb{P}^{n+1}\times Gr(1,m+1), \ell \cdot X \geq (b-c)x_{t}\} $. Moreover, the gonality of the curve $f(\mathcal{C}_{t})$ is computed by projecting from the point $x_{t}$.
\end{prop}

\begin{thm}[{\cite[Theorem 4.1]{bastianelli2019gonality}}]
\label{Bastianelli}
Let $X \subset \mathbb{P}^{n+1}$ be a very general irreducible smooth hypersurface of degree $d\geq 2n+2$. Then:
\begin{center}
 $ d-\lfloor \frac{\sqrt{16n+9}-1}{2} \rfloor \leq \covgon(X)\leq d-\lfloor \frac{\sqrt{16n+1}-1}{2} \rfloor $.
\end{center}

Let $X \subset \mathbb{P}^{n+1}\times \mathbb{P}^{m+1}$ with $m,n \geq 2$ be a very general irreducible smooth hypersurface of bi-degree $(a,b)$, such that $a \geq 2n+m+1,b\geq 2m+n+1$. Then:
\begin{center}
 ${ \minimum \Big\{a-\lfloor \frac{\sqrt{16n+9}-1}{2} \rfloor,b-\lfloor \frac{\sqrt{16m+9}-1}{2} \rfloor \Big\} \leq \covgon(X)\leq \minimum \Big\{a-\lfloor \frac{\sqrt{16n+1}-1}{2} \rfloor,b-\lfloor \frac{\sqrt{16m+1}-1}{2} \rfloor \Big\} }$.
\end{center}
Moreover, both sides of the inequality coincide whenever $m,n \in \{4\alpha^{2}+3\alpha, 4\alpha^{2}+5\alpha+1, \alpha \in \mathbb{N}\}$
\end{thm}

\begin{proof}
As stated throughout this work, $X$ can alternatively be seen as a family of hypersurfaces of degree $a$ over $\mathbb{P}^{m+1}$ or as a family of hypersurfaces of degree $b$ over $\mathbb{P}^{n+1}$. Proposition \ref{LocalisationOfCurves} implies that $f(C_{t})$ is contained in $V_{x_{t}}^{a-c}$ for a certain $x_{t} \in X $ or in $ W^{b-c}_{x_{t}}$. Suppose, for instance that $f(C_{t}) \subset V_{x_{t}}^{a-c}$. More precisely, as a line that is in $V_{x_{t}}^{a-c}$ is in $Gr(1,n+1)\times \pi_{2}(x_{t})$, this shows that for a general $t\in T$ we have $f(\mathcal{C}_{t})\subset \pi^{-1}_{2}(\pi_{2}(x_{t}))$, i.e. the curve is on one of the hypersurfaces of the family. This means, that computing $\covgon(X)$ can be done fibrewise. Using Theorem \ref{Bastianelli}, this implies that $\covgon(X)\geq \minimum \Big\{a-\lfloor \frac{\sqrt{16n+9}-1}{2} \rfloor,b-\lfloor \frac{\sqrt{16m+9}-1}{2} \rfloor \Big\}$.
Moreover, the upper bound is clear as one can construct a family of curves on $X$ by constructing it fibrewise using the family of curves mentioned in Theorem \ref{Bastianelli}. This yields the desired upper bound. As in \ref{Bastianelli}, both side of the inquality coincide whenever Moreover, both sides of the inequality coincide whenever $m,n \in \{4\alpha^{2}+3\alpha, 4\alpha^{2}+5\alpha+1, \alpha \in \mathbb{N}\}$.
\end{proof}
\begin{rmq}
We would like to make some comments on the assumption on the range of the bi-degree $(a,b)$ of the hypersurfaces $X$ of our theorem. Indeed, as already stated before, the assumptions $a\geq 2m+n+1, b\geq 2n+m+1$ are only required to get the lower bound on $\covgon(X)$; while the upper bound is achieved as soon as $a \geq 2n+2, b\geq 2m+2$. This was required to carry out the computations made during the proof of the Proposition \ref{LocalisationOfCurves}. Following \cite{yeong2022algebraic}, these numerical assumptions are actually the ones ensuring that $X$ is algebraically hyperbolic, otherwise, Yeong shows that $X$ contains a line. As the proof relied repeatedly on the fact that $X$ does not contain any line, one would need to change the method of the proof to possibly lower the bound on $a$ and $b$.
\end{rmq}

\vspace{0.35cm}
First, this computes the covering gonality of such hypersurfaces up to one, in the same spirit as the result of \cite{bastianelli2019gonality}. This result admits a geometric interpretation, which lies at the core of this work. Consider $X \subset \mathbb{P}^{n+1} \times \mathbb{P}^{m+1}$ a very general smooth irreducible of bi-degree $(a,b)$ with $a \geq 2n+m+1, b \geq 2m+n+1$. Via the natural projection onto each factor, $X$ may alternatively be seen as a family of degree $a$ hypersurfaces in $\mathbb{P}^{n+1}$ parametrized by $\mathbb{P}^{m+1}$, or as a family of hypersurfaces of degree $b$ in $\mathbb{P}^{m+1}$ parametrized by $\mathbb{P}^{n+1}$. We prove that the covering gonality of $X$ is actually the covering gonality of a general fibre from one of these families. \\
In the final section of this paper we study the newly introduced birational invariant measure of association. This notion has been introduced by Lazarsfeld-Martin in \cite{lazarsfeld2023measures} and the intuition is as follow: given a pair of two algebraic variety of the same dimension $n$, one can be interested by the following question: are $X$ and $Y$ birationally related? If not how can one measure that failure? In \cite{lazarsfeld2023measures} Lazarfeld and Martin introduce the joint covering gonality of two algebraic varieties of same dimension $X$ and $Y$, that we denote $\covgon(X,Y)$. Although less understood than in the classical setting, this invariant already satisfies interesting lower bounds. Our second main result establishes such a bound for hypersurfaces in products of projective spaces inspired by the one proven in \cite{lazarsfeld2023measures}.

\begin{thm}
Let $X, Y \subset \mathbb{P}^{n+1} \times \mathbb{P}^{m+1}$ be very general smooth hypersurfaces
of bi-degrees $(a_1,b_1)$ and $(a_2,b_2)$ respectively with $a_{1},a_{2} \geq 2n+m+2$ and $b_{1},b_{2} \geq 2m+n+2$. Then:
\vspace{0.5cm}
\begin{center}
   $\operatorname{cov.gon}(X,Y) \;\geq\; k_1 + k_2 - n - m$
\end{center}
where :
\[
k_1 = \min\{a_1 - n - 3, b_1 - m - 3\}, \quad
k_2 = \min\{a_2 - n - 3, b_2 - m - 3\}.
\]

\end{thm}

Concerning the organisation of this paper it will be as follows. In the first section we recall all the needed background on the covering gonality and the classical way one has to control this invariant. The second section, much shorter is devoted to define the notion of line in a product of projective spaces which we use heavily during the proof of the first theorem in Section 3. Finally, in section 4 we explore the joint covering gonality and prove the second main theorem stated above.
\\

 In this work, $X$ will denote a smooth irreducible  complex projective variety of  dimension $\dim(X)$. We will say a property holds at a general point $x \in X $ if it holds on a Zariski open subset of $X$. We will say that a property holds for a very general point in $X$ if it holds outside of countably many proper Zariski closed subsets in $X$.

\vspace{0.3cm}

\textit{Acknowledgments}:
I would first like to express my sincere gratitude to my advisors, Damian Brotbek and Gianluca Pacienza, for their time, guidance, and valuable feedback on this work and its earlier versions. This paper owes a great deal to Francesco Bastianelli, who delivered a series of lectures at the IECL during which he generously proposed this problem. The present work is deeply inspired by his research. Finally, I warmly thank Steve Balme for his encouragement throughout the development of this work.

\begin{center}

We prove that the covering gonality of $X$ is actually the covering gonality of a general fibre from one of these families: via the natural projections $X\to \mathbb{P}^{m+1}$ and $X\to \mathbb{P}^{n+1}$, one can view $X$ either as “a family of degree $a$ hypersurfaces in $\mathbb{P}^{n+1}$ parametrized by $\mathbb{P}^{m+1}$” or as “a family of hypersurfaces of degree $b$ in $\mathbb{P}^{m+1}$ parametrized by $\mathbb{P}^{n+1}$,” and the argument identifies $\covgon(X)$ with the covering gonality of a general fibre from one of these two families.

\label{mainthm}
Let $X \subset \mathbb{P}^{n+1}\times \mathbb{P}^{m+1}$ be a very general irreducible smooth hypersurface of bi-degree $(a,b)$. Suppose that $a\geq 2n+m+1$, $b\geq 2m+n+1$, and $n,m \geq 2 $. Then:
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    ${ \minimum \Big\{a-\lfloor \frac{\sqrt{16n+9}-1}{2} \rfloor,b-\lfloor \frac{\sqrt{16m+9}-1}{2} \rfloor \Big\} \leq \covgon(X)\leq \minimum \Big\{a-\lfloor \frac{\sqrt{16n+1}-1}{2} \rfloor,b-\lfloor \frac{\sqrt{16m+1}-1}{2} \rfloor \Big\} }$.
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Moreover, both sides of the inequality coincide whenever $m,n \in \{4\alpha^{2}+3\alpha, 4\alpha^{2}+5\alpha+1, \alpha \in \mathbb{N}\}$,

Question: Let \(X \subset \mathbb{P}^{n+1}\times \mathbb{P}^{m+1}\) be a very general irreducible smooth hypersurface of bi-degree \((a,b)\), where \(n,m\ge 2\), \(a\ge 2n+m+1\), and \(b\ge 2m+n+1\). Here \(\operatorname{cov.gon}(X)=\covgon(X)\) denotes the covering gonality of \(X\), i.e. the minimal gonality of an irreducible curve passing through a general point of \(X\), and the gonality of a smooth irreducible curve \(C\) is the minimal degree of a morphism \(C\to \mathbb{P}^1\). Which statement holds for every such hypersurface \(X\)?

\[\min\!\left\{a-\left\lfloor \frac{\sqrt{16n+9}-1}{2}\right\rfloor,\; b-\left\lfloor \frac{\sqrt{16m+9}-1}{2}\right\rfloor\right\}\le \covgon(X)\le \min\!\left\{a-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor,\; b-\left\lfloor \frac{\sqrt{16m+1}-1}{2}\right\rfloor\right\}.\] Moreover, if both \(m\) and \(n\) belong to \(\{4\alpha^2+3\alpha,\;4\alpha^2+5\alpha+1\mid \alpha\in\mathbb{N}\}\), then the lower and upper bounds above coincide.

Choices:
B. \[\min\!\left\{a-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor,\; b-\left\lfloor \frac{\sqrt{16m+1}-1}{2}\right\rfloor\right\}\le \covgon(X)\le \min\!\left\{a-\left\lfloor \frac{\sqrt{16n+9}-1}{2}\right\rfloor,\; b-\left\lfloor \frac{\sqrt{16m+9}-1}{2}\right\rfloor\right\}.\] Moreover, if both \(m\) and \(n\) belong to \(\{4\alpha^2+3\alpha,\;4\alpha^2+5\alpha+1\mid \alpha\in\mathbb{N}\}\), then the lower and upper bounds above coincide.
C. \[\covgon(X)\ge \min\!\left\{a-\left\lfloor \frac{\sqrt{16n+9}-1}{2}\right\rfloor,\; b-\left\lfloor \frac{\sqrt{16m+9}-1}{2}\right\rfloor\right\}.\]
D. \[\max\!\left\{a-\left\lfloor \frac{\sqrt{16n+9}-1}{2}\right\rfloor,\; b-\left\lfloor \frac{\sqrt{16m+9}-1}{2}\right\rfloor\right\}\le \covgon(X)\le \max\!\left\{a-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor,\; b-\left\lfloor \frac{\sqrt{16m+1}-1}{2}\right\rfloor\right\}.\] Moreover, if both \(m\) and \(n\) belong to \(\{4\alpha^2+3\alpha,\;4\alpha^2+5\alpha+1\mid \alpha\in\mathbb{N}\}\), then the lower and upper bounds above coincide.
E. \[\min\!\left\{a-\left\lfloor \frac{\sqrt{16n+9}-1}{2}\right\rfloor,\; b-\left\lfloor \frac{\sqrt{16m+9}-1}{2}\right\rfloor\right\}\le \covgon(X)\le \min\!\left\{a-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor,\; b-\left\lfloor \frac{\sqrt{16m+1}-1}{2}\right\rfloor\right\}.\] Moreover, the lower and upper bounds above coincide for every such pair \((m,n)\) with \(m,n\ge 2\).

assignment of fibrewise lower versus upper bounds

dropped the upper bound and coincidence clause, keeping only the fibrewise lower bound

minimum over the two projection families replaced by maximum

restricted coincidence condition replaced by uniform coincidence for all dimensions

The stem does not reveal the correct bound; it only states the hypotheses and asks for the quantitative estimate. There is no explicit answer leakage, though the setup clearly signals that a theorem-level statement is being recalled.

This is essentially a direct theorem-recall item: the stem gives the full hypotheses and asks for the exact conclusion. The correct option is basically the theorem statement itself rather than a derived consequence or an application.

There is some moderate reasoning in checking which floor-shift belongs to the lower versus upper bound, whether a min or max is appropriate, and whether the equality condition should require both indices. However, the item mainly tests recognition/recall of a precise statement rather than genuine generative mathematical reasoning.

The distractors are mathematically plausible and target realistic failure modes: reversing lower/upper bounds, replacing min by max, weakening the equality criterion, and offering a weaker true statement. They are distinct and nontrivial.

A solid theorem-recognition MCQ with no answer leakage and strong distractors, but it is largely tautological and only moderately tests reasoning rather than generation.