  The MinRank (MR) problem is a computational problem that arises in many
cryptographic applications. In Verbel et al. (PQCrypto 2019), the authors
introduced a new way to solve superdetermined instances of the MinRank problem,
starting from the bilinear Kipnis-Shamir (KS) modeling. They use linear algebra
on specific Macaulay matrices, considering only multiples of the initial
equations by one block of variables, the so called ''kernel'' variables. Later,
Bardet et al. (Asiacrypt 2020) introduced a new Support Minors modeling (SM),
that consider the Pl{\"u}cker coordinates associated to the kernel variables,
i.e. the maximal minors of the Kernel matrix in the KS modeling. In this paper,
we give a complete algebraic explanation of the link between the (KS) and (SM)
modelings (for any instance). We then show that superdetermined MinRank
instances can be seen as easy instances of the SM modeling. In particular, we
show that performing computation at the smallest possible degree (the ''first
degree fall'') and the smallest possible number of variables is not always the
best strategy. We give complexity estimates of the attack for generic random
instances.We apply those results to the DAGS cryptosystem, that was submitted
to the first round of the NIST standardization process. We show that the
algebraic attack from Barelli and Couvreur (Asiacrypt 2018), improved in Bardet
et al. (CBC 2019), is a particular superdetermined MinRank instance.Here, the
instances are not generic, but we show that it is possible to analyse the
particular instances from DAGS and provide a way toselect the optimal
parameters (number of shortened positions) to solve a particular instance.
