In this paper under review, the authors study the space of unicuspidal rational curves of degree \(d\) and (arithmetic) genus \(g\) in \(\mathbb{P}^n\), i.e. curves whose singularities are unibranch singletones. Inside \(M^n_d\) which is defined as the space of non-degenerate morphisms \(f:\mathbb{P}^1\rightarrow\mathbb{P}^n\) of degree \(d>0\), one considers the subvariety \(M^n_{d,g}\) of morphisms whose images have arithmetic genus \(g>0\), which are necessarily singular. If \(P\in C=f(\mathbb{P}^1)\) is a unibranch singularity i.e. a cusp, one may associate a value semigroup \(S\) of \(P\) as follows. Denoting by \(\psi :t\mapsto (\psi_1(t),\cdots,\psi_n(t))\) a local parametrization at \(P\) corresponding to a map of rings \(\phi: R=\mathbb{C}[[x_1,\cdots,x_n]]\rightarrow\mathbb{C}[[t]]\), \(S:=\nu(\phi(R))\) is the value semigroup of \(P\) where \(\nu\) is the standard valuation.
In this paper, the authors take a closer look at rational unicuspidal curves whose value semigroup \(S\) is \(\gamma\)-hyperelliptic, i.e. when \(S=<4, 4\gamma + 2, 2g-4\gamma + 1>\). When \(\gamma =0\), it is proved in Theorem 1 that the subvariety of rational curves with a unique cuspidal singularity with valuation semigroup \(S=<2,2g+1>\) is codimension at least \((n-1)g\) in \(M^n_d\). To prove the result the authors produce an explicit packet of polynomials in the \(i\)-th component \(f_i's\) – defining the morphism \(f:\mathbb{P}^1\rightarrow\mathbb{P}^n\)- that impose independent conditions on their coefficients. They also showed that this bound is sharp for small \(g\); Proposition 2.5.
They also treated rational curves with \(\gamma\)-hyperelliptic cusps of maximal weight. In Theorem 3.1, when \(g>>\gamma\) they obtain an explicit lower bound for the codimension of rational curves with \(\gamma\)-hyperelliptic cusps of maximal weight.