Weierstrass semigroups and nodal curves of type \(p,q\) (Q411772)

Let \(\mathcal{R}\) be a smooth projective curve of genus \(g\) over an algebraically closed field \(K\) of characteristic \(0\). The Weierstraß semigroup \(H(P)\subset\mathbb{N}\) is the semigroup of all possible pole orders of meromorphic functions on \(\mathcal{R}\) which at worst have a pole at \(P\). The set of gaps of \(H(P)\) can be described by the holomorphic differentials \(\omega\) on \(\mathcal{R}\): NEWLINE\[NEWLINE\mathbb{N}\setminus H(P):=\{\mathrm{ord}_P(\omega)+1\mid \omega\in H^0(K_{\mathcal{R}})\}.NEWLINE\]NEWLINE This paper under review studies the Weierstraß semigroups of \(\mathcal{R}\) by investigating their plane models of type \(p, q\) with nodal singularities. A plane curve of type \(p, q\) is defined by equation NEWLINE\[NEWLINEY^p+aX^q+\sum_{\nu p+\mu q<pq}a_{\nu\mu}X^{\nu}Y^{\mu}=0,NEWLINE\]NEWLINE where \(1<p<q\), \(p\), \(q\) coprime. Such curves \(C\) are irreducible and have only one place \(P\) at infinity which is a point on the normalization \(\mathcal{R}\) of the projective closure of \(C\). We also call \(H(P)\) the Weierstraß semigroup of \(C\). There are two main results proved in this paper.NEWLINENEWLINEThe first result (Theorem 6.4) is that for any \(\mathcal{R}\) and \(H(P)\), \(\mathcal{R}\) has a plane model of type \(p, q\) with the place \(P\) at infinity which has at worst nodes as singularities for suitable \(p\) and \(q\). In particular every Weierstraß semigroup is the Weierstraß semigroup of a plane nodal curve of type \(p, q\). Therefore one can focus on studying the Weierstraß semigroup of plane nodal curves of type \(p, q\).NEWLINENEWLINEThe second result (Proposition 4.2) computes the gaps of \(H(P)\) (therefore \(H(P)\)) for any such plane nodal curve provided the nodes are explicitly known.NEWLINENEWLINEAs applications, the authors construct many plane curves of type \(p, q\) with explicitly given nodes and determine their Weierstraß semigroups.