It is well known that the existence of projectively canonical algebraic surfaces of given degree $n$ of $\pn^3(\comp)$, that is the surfaces of $\pn^3(\comp)$ whose prime sections are the canonical curves, requires the existence of two surfaces $F$, $G$ of degree $n$ and $n-5$ tangent to each other along a curve of degree $\frac{1}{2}n(n-5)$ passing trough $\left (\begin{array}{c} n-4\\ 3\end{array} \right )$ double points of $G$. We prove that this contact situation is possible at least up to $n\leq 20$. Incidentally, we propose very short and natural proofs of some known results (see [4], [5]) and hint at a new strategy for obtain canonical surfaces.
On regular canonical surfaces of genus four
CANONERO, GABRIELLA;GALLARATI, DIONISIO;SERPICO, MARIA EZIA
2006-01-01
Abstract
It is well known that the existence of projectively canonical algebraic surfaces of given degree $n$ of $\pn^3(\comp)$, that is the surfaces of $\pn^3(\comp)$ whose prime sections are the canonical curves, requires the existence of two surfaces $F$, $G$ of degree $n$ and $n-5$ tangent to each other along a curve of degree $\frac{1}{2}n(n-5)$ passing trough $\left (\begin{array}{c} n-4\\ 3\end{array} \right )$ double points of $G$. We prove that this contact situation is possible at least up to $n\leq 20$. Incidentally, we propose very short and natural proofs of some known results (see [4], [5]) and hint at a new strategy for obtain canonical surfaces.
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