Non-general type surfaces in \(\mathbb{P}^4\): Some remarks on bounds and constructions
The authors focus on one specific problem of classification of non-general type surfaces in \(\mathbb{P}^4\). Via linear projection of a general point outside the secant variety of a surface \( S \subset \mathbb{P}^n \), \(S\) can be embedded in \(\mathbb{P}^5\). Due to the classical result of \textit{F. Severi} [Rend. Circ. Math. Palermo 15, 33-51 (1901; JFM 32.0648.04)] which states that if the secant variety of \(S\) is not all \(\mathbb{P}^5\) then \(S\) is the Veronese surface in \(\mathbb{P}^5\), one expects constraints on the numerical invariants of a smooth surface in \(\mathbb{P}^4\), which satisfy the double point formula. In this direction \textit{G. Ellingsrund} and \textit{C. Peskine} [Invent. Math. 95, No. 1, 1-11 (1989; Zbl 0676.14009)] have shown that there exists an integer \(d_0\) such that every smooth surface in \(\mathbb{P}^4\) of degree \(d>d_0\) is of general type. The authors discuss various bounds for \(d_0\) already known. For example Ellingsrud and Peskine [loc.cit] have shown that \(d_0 \leq 10.000\), with arguments by \textit{R. Braun} and \textit{G. Fløystad} [Compos. Math. 93, No. 2, 211-229 (1994; Zbl 0823.14021)] the authors show that \(d_0 \leq 105\), and \textit{M. Cook} [Compos. Math. 102, No. 2, 141-145 (1996; Zbl 0871.14033)] showed that \(d_0 \leq 76\). The authors prove the following theorem: Let \(S \subset \mathbb{P}^4\) be a smooth surface which is not of general type. Then the degree of \(S\) is smaller or equal to 52. In the first part of their paper they discuss the degree bound. In the second part of the paper the authors explain the method by \textit{W. Decker, L. Ein} and \textit{F.-O. Schreyer} [J. Algebr. Geom. 2, No. 2, 185-237 (1993; Zbl 0795.14019)] which relies on symbolic computation, adjunction theory and the approach of \textit{S. Popescu} and \textit{K. Ranestad} [J. Algebr. Geom. 5, No. 1, 13-76 (1996; Zbl 0861.14041)] of classification.
classification of non-general type surfaces
computational aspects in algebraic geometry
scientific article; zbMATH DE number 1471046
scientific article