Commentarii Mathematici Helvetici


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Volume 91, Issue 3, 2016, pp. 477–491
DOI: 10.4171/CMH/392

Explicit Brill–Noether–Petri general curves

Enrico Arbarello[1], Andrea Bruno, Gavril Farkas[2] and Giulia Saccà[3]

(1) Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale A. Moro 2, 00185, Roma, Italy
(2) Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Raum 1.401, 10099, Berlin, Germany
(3) Department of Mathematics, Stony Brook University, NY 11794-3651, Stony Brook, USA

Let $p_1,\dots, p_9$ be the points in $\mathbb A^2(\mathbb Q)\subset \mathbb P^2(\mathbb Q)$ with coordinates $$(-2,3),(-1,-4),(2,5),(4,9),(52,375), (5234, 37866),(8, -23), (43, 282), \Bigl(\frac{1}{4}, -\frac{33}{8} \Bigr),$$ respectively. We prove that, for any genus $g$, a plane curve of degree $3g$ having a $g$-tuple point at $p_1,\dots, p_8$, and a $(g-1)$-tuple point at $p_9$, and no other singularities, exists and that the general plane curve of that degree and with those singularities is a Brill–Noether–Petri general curve of genus $g$.

Keywords: Brill–Noether theory, moduli of curves, surfaces with canonical sections

Arbarello Enrico, Bruno Andrea, Farkas Gavril, Saccà Giulia: Explicit Brill–Noether–Petri general curves. Comment. Math. Helv. 91 (2016), 477-491. doi: 10.4171/CMH/392