Two remarks on the Weierstrass flag (Q2895544)

In the paper under review the authors consider the Weierstrass flag introduced by \textit{E. Arbarello} [Compos. Math. 29, 325--342 (1974; Zbl 0355.14013)]. Namely, if \(M_g\) denotes the quasiprojective variety of smooth curves of genus \(g\), and \(M_{g,1}\) denotes the variety of smooth curves with one marked point, they consider NEWLINE\[NEWLINE \overline{W}_* (2) \subset \overline{W}_* (3) \subset \dots \subset \overline{W}_* (g+1) = M_{g,1} \, NEWLINE\]NEWLINE where \(\overline{W}_* (d) = \{ [(C,p)] \in M_{g,1} \, | \, h^0 (C, dp) \geq 2 \}\). Remark that if \(\pi : M_{g,1} \to M_g\) is the forgetful map, then \(\pi (\overline{W}_* (d) ) = \overline{W} (d) = \{ [C] \in M_g \mid \exists p \in C \text{ such that } h^0 (C, dp) \geq 2 \}\), and NEWLINE\[NEWLINE\overline{W} (2) \subset \overline{W} (3) \subset \dots \subset \overline{W}_* (g) = M_gNEWLINE\]NEWLINE is a filtration of \(M_g\). The first result of the authors, Proposition 3.1, is about NEWLINE\[NEWLINEW_* (d) = \overline{W}_* (d) \setminus \overline{W}_* (d-1) = \{ [(C,p)] \in M_{g,1} \mid h^0 (C, dp) = 2 \} ,NEWLINE\]NEWLINE as they prove that if \(5 \leq d \leq g+1\) and \(d\) is not prime, or \(d\) is prime but \((d-1)(d-2) \geq 2g\), then \(W_* (d)\) couldn't be affine. The idea of the proof is to show that the strata can be realized as open subsets of smooth varieties, whose complement (representing a family of plane curves of degree \(d\) with a point at which the tangent intersects with multiplicity \(d\)) is not purely divisorial. The second result, Proposition 3.2, is about the locally closed strata NEWLINE\[NEWLINE\begin{multlined} W (d) = \overline{W} (d) \setminus \overline{W} (d-1) = \{ [C] \in M_g \mid \exists p \in C \\ \text{ such that } h^0 (C, dp) = 2 , \text{ and } h^0 (C, (d-1)q) = 1 \text{ for } \forall q \in C \} .\end{multlined}NEWLINE\]NEWLINE They prove that if \(g \geq 6\) and \(5 \leq d \leq g-1\), then \(W (d)\) couldn't be affine either. The argument is similar, as the authors apply a small extension of a result of \textit{S. Diaz} [J. Differ. Geom. 20, 471--478 (1984; Zbl 0565.14011)], about the dimension of locus of curves with two exceptional Weierstrass points.NEWLINENEWLINEFor the entire collection see [Zbl 1236.14001].