On the projective normality of Artin-Schreier curves (Q2868756)

Artin-Schreier curves are plane curves defined over a field \(\mathbb F\) of characteristic \(p\) by an equation of the form NEWLINE\[NEWLINE Y_f: y^q+y=f(x), NEWLINE\]NEWLINE where \(q\) is a power of \(p\) and \(f(x)\in \mathbb F[x]\) is a polynomial of degree \(m\) with \((m,p)=1\). Many of these curves are singular and their genus, defined as the genus of their normalization \(C_f\), is \((m-1)(q-1)/2\).NEWLINENEWLINEIn this paper, the authors study the projective normality of certain embeddings \(X_f\) of the curves \(C_f\) into suitable projective spaces. In the case \(m\mid (q-1)\), if \((q-1)/m =1\) the curve \(Y_f\) is a smooth plane curve, while if \(c=(q-1)/m > 1\) the authors prove that \(X_f\subseteq \mathbb P^{c+1}\) and that it is projectively normal, being contained, when \(m\geq 3\), within \((c+3)(c+2)/2 - 3c -3\) linearly independent quadric hypersurfaces. The authors also study the case \(m\mid (tq+1)\), for integers \(t\geq 1\) and curves with \(f(x)=x^m\). If \((tq+1)/m \leq q-1\) they prove that \(X_f\) is projectively normal as well.