On the Chern numbers of surfaces and 3-folds of codimension 2 (Q675762)

The author's aim is to give results concerning the slopes, i.e. the ratios of Chern numbers of surfaces in \(\mathbb{P}^4\) and 3-folds in \(\mathbb{P}^5\). There are many results in this direction, in particular Miyaoka's inequality and Sommese's theorem which shows that any rational number in \([1/5,3]\) occurs as the slope \(c_1^2/c_2\) of a surface of general type. The main results of the author are the following: The slopes of determinantal surfaces of general type are all rational numbers between 1 and \(7/5\) and an infinite sequence accumulating to 1 from below. All rational numbers between 1 and \(5/3\) are slopes of surfaces in \(\mathbb{P}^4\). Concerning 3-folds in \(\mathbb{P}^5\), the author shows that the set of slopes \(c_1^3/(c_1 c_2)\) of determinantal 3-folds in \(\mathbb{P}^5\) consists of all rational numbers between 1 and \(17/12\) and an infinite sequence accumulating from below. The same result is obtained for the slopes \(c_1^3/c_3\) with upper bound \(17/7\).