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# Commentarii Mathematici Helvetici

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**Volume 91, Issue 3, 2016, pp. 477–491**

**DOI: 10.4171/CMH/392**

Published online: 2016-08-18

Explicit Brill–Noether–Petri general curves

Enrico Arbarello^{[1]}, Andrea Bruno, Gavril Farkas

^{[2]}and Giulia Saccà

^{[3]}(1) Università di Roma La Sapienza, Italy

(2) Humboldt-Universität zu Berlin, Germany

(3) Stony Brook University, 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