Let us call a class $\calP$ of posets iterable, if, for any $\poP\in\calP$ and $\calP$-name

$\utpoQ$\vspace{-0.5\smallskipamount} \st\ $\forces{\poP}{\utpoQ\in\calP}$, we have

$\poP\ast\utpoQ\in\calP$.

For an iterable class $\mathcal{P}$ of posets, a cardinal $\mu$ is called {\it Laver-generically

supercompact for $\mathcal{P}$}, if, for any $\mathbb{P}\in\mathcal{P}$ and $\lambda\in\On$,

there is a $\poP$-name $\utpoQ$\vspace{-0.5\smallskipamount} with $\forces{\poP}{\utpoQ\in\calP}$ \st, letting

$\poQ=\poP\ast\utpoQ$,

there are $j$, $M\subseteq\uniV[\genH]$ for $(\uniV, \mathbb{Q})$-generic

$\genH$ such that\\

\\

\assert{1} $\elembed{j}{V}{M}$,\smallskip

\\

\assert{2} $crit(j)=\mu$, $j(\mu)>\lambda$,\smallskip

\\

\assert{3} $\cardof{\poQ}\leq j(\mu)$,\smallskip

\\

\assert{4} $\poP$, $\genH\in M$ and \smallskip

\\

\assert{5} $j\imageof\lambda\in M$.\\\\

The notion of Laver-generically superhugeness is obtained when \assert{5} is replaced by\\

\\

\assert{5'} $j\imageof j(\mu)\in M$.

\\

The notion of Laver-generically large cardinal for $\calP$ given here is stronger than the one

introduced in \cite{II} and is called there the {\it strongly} and {\it tightly}

Laver-generically large cardinal (the strongness corresponds the usage of two-step

iteration in the definition instead of just $\poP\circleq\poQ$, and the tightness the

condition \assert{3}).

In my talk, I will give a proof of the following:\quad

For many natural iterable class of proper posets $\mathcal{P}$, a

Laver-generically supercompact cardinal $\mu$ for $\poP$ is either $\aleph_2$ or very large (if it

exists),

and the continuum is either $\aleph_1$ or $\aleph_2$, or $\geq\mu$ in case of very large

$\mu$, where it depends on $P$ which senario we have.

If time allows, I will also sketch a proof of the following theorem:\quad

If $\mathcal{P}$ is the class of c.c.c.\ posets (or some other iterable class $\calP$ of posets preserving all

cardinalities but adding some real), and if $\mu$ is Laver-generically superhuge for $\mathcal{P}$, then

$\mu=2^{\aleph_0}$.

At the moment, it is open if the same theorem holds for a Laver-generically supercompact

cardinal.

\begin{thebibliography}{xxx}

\bibitem{II} S.F., Andr\'e Ottenbreit Maschio Rodrigues and Hiroshi Sakai, Strong

downward L\"owenheim-Skolem theorems for stationary logics, II --- \\reflection down to

the continuum, to appear in Archive for Mathematical Logic (202?).\quad

\scalebox{0.9}[1]{\color{blue}\tt https://fuchino.ddo.jp/papers/SDLS-II-x.pdf}

\end{thebibliography}