Surface counterexamples to the Eisenbud-Goto conjecture (Q6571611)

The Eisenbud-Goto Conjecture states that for any projective variety \(X\) one has\N\[\N\operatorname{reg}X\leq \deg X-\operatorname{codim}X+1.\N\]\N\N\textit{J. McCullough} and \textit{I. Peeva} [J. Am. Math. Soc. 31, No. 2, 473--496 (2018; Zbl 1390.13043)] showed that there are counterexamples to the Eisenbud-Goto conjecture, proving that the regularity of projective varieties is not bounded by any polynomial function of the degree.\N\NIn spite of the counterexamples, the Eisenbud-Goto conjecture is still believed to hold for projective varieties with nice properties. Also as J. McCullough and I. Peeva stated, the following are interesting cases where the conjecture may hold: smooth projective varieties, projectively normal varieties, projective toric varieties, and projective singular surfaces.\N\NHowever, in this paper, the authors construct counterexamples of projective singular surfaces of codimension two to the Eisenbud-Goto conjecture. At the moment, these are the first surface counterexamples to the conjecture.\N\NThe counterexamples are constructed via binomial rational maps between projective spaces and are based on the following\N\N{Theorem A.} Let \(m \geq 6\) be an integer and \(X_m\) be the image of the binomial rational map\N\[\N\varphi_m:\mathbb{P}^2=\operatorname{Proj}\mathbb{K}[y_0,y_1,y_2]\dashrightarrow \mathbb{P}^4=\operatorname{Proj}\mathbb{K}[x_0,\dots, x_4]\N\]\Ngiven by the linear system\N\[\N(\mathcal{O}_{\mathbb{P}^2}(m),y_1^{m-1}y_2,y_0^{m-1}y_1,y_1^m+y_1^{m-3} y_2^3, y_0^m,y_2^{m-1}y_0).\N\]\NThen \(X_m\subset \mathbb{P}^4\) is a surface of degree \(m^2-m+3\). In addition, suppose \(m = 6k\) for some integer \(k \geq 1\). If \(L(k)\) and \(W(k)\) are both invertible matrices, then \(\operatorname{maxdeg}Xm \geq \frac{3}{2}m^2 - \frac{7}{2}m+1\). In this case, the surface \(X_m\subset \mathbb{P}^4\) is a counterexample of the Eisenbud-Goto conjecture.\N\NHere, \(L(k)\) and \(W(k)\) are the matrices whose rows correspond to some polynomials to consider. Moreover, for a homogeneous ideal \(I\), the largest degree of an element in a minimal generating set of \(I\) is denoted as \(\operatorname{maxdeg} I\), which is less or equal to \(\operatorname{reg} I\). For a projective variety \(X\) with homogeneous ideal \(I_X\), let us define \(\operatorname{maxdeg}X := \operatorname{maxdeg} I_X\).\N\NA seocond important result of the paper is the following\N\N{Theorem B.} The surface \(X_6 \subseteq \mathbb{P}^4\) is a counterexample to the Eisenbud-Goto conjecture as \(\operatorname{maxdeg}X_6 \geq 34\) and deg \(X_6 = 33\). It violates the normality conjecture as it is neither 31-normal nor 32-normal.