Let \(C\) be a smooth nonhyperelliptic curve of genus \(g > 2\). It is called \(s\)-subcanonical if there exists a very ample line bundle \(L\) on \(C\) such that \(L^{\otimes s} = \omega_C\), where \(\omega_C\) is the canonical line bundle on \(C\). Consider the embedding \(C \subset {\mathbb P}^r\) induced by \(L\), where \(r = \dim H^0 (C, L) - 1\). The line bundle \(L\) is called normally generated if for the above embedding \(H^0 ({\mathbb P}^r , O_{P^r} (k)) \to H^0 (C , L^{\otimes k})\) is surjective for every \(k \geq 1\), i.e. the embedded curve is projectively normal. The minimal integer \(k\) for which there exists a line bundle of degree \(k\) on \(C\) with a non-trivial global section is called gonality and \(C\) is referred as \(k\)-gonal curve.
The paper under review deals with the problem of identifying the \(s\)-subcanonical \((s+2)\)-gonal projectively normal curves. The idea is to consider the homogeneous coordinate ring of the embedded curve, say \(S_C = \bigoplus^{\infty}_{k = 0} H^0 (C, L^{\otimes k})\), and two general linear forms \(\eta_1, \eta_2 \in S_C\). Then \(A = S_C / \langle \eta_1, \eta_2 \rangle\) turns out to be an Artinian Gorenstein ring of socle dimension one and degree \(s+2\). Via the Macaulay correspondence, such a ring can be realized as the apolar ring of some homogeneous polynomial \(F \in {\mathbb C} [x_0 , \dots , x_{r-2}]\) of degree \(s+2\), i.e. \(A = {\mathbb C} [x_0 , \dots , x_{r-2}] / F^{\bot}\), where \(F^{\bot} = \{ D \in {\mathbb C} [\partial_0 , \dots , \partial_{r-2}] \;| \;D(F) = 0 \}\) with \({\mathbb C} [\partial_0 , \dots , \partial_{r-2} ]\) being the ring of natural derivations over \({\mathbb C} [x_0 , \dots , x_{r-2}]\).
Thus, one can relate geometric properties of a projectively normal \(s\)-subcanonical curve \(C\) like its gonality, or existence of smooth plane model of it with algebraic properties of \(F\), and more precisely whether it can be represented as \(F = l^{s+2}_0 + \dots + l^{s+2}_k\) (Fermat hypersurface), where \(l_0, \dots , l_k\) are linearly independent forms of degree one. Note that the minimal such \(k\) is called the Waring number of \(F\). It turns out that this is exactly the case, as the authors prove that all projectively normal \(s\)-subcanonical curves for which \(F\) has degree \(s+2\) are either \(s+2\) gonal or have smooth plane model of degree \(2s+3\).