We propose here a generalization of the problem addressed by the SHGH conjecture. The SHGH conjecture posits a solution to the question of how many conditions a general union $X$ of fat points imposes on the complete linear system of curves in $\mathbb{P}^{2}$ of fixed degree $d$, in terms of the occurrence of certain rational curves in the base locus of the linear subsystem defined by $X$. As a first step towards a new theory, we show that rational curves play a similar role in a special case of a generalized problem, which asks how many conditions are imposed by a general union of fat points on linear subsystems defined by imposed base points. Moreover, motivated by work of Di Gennaro, Ilardi and Vallès and of Faenzi and Vallès, we relate our results to the failure of a strong Lefschetz property, and we give a Lefschetz-like criterion for Terao’s conjecture on the freeness of line arrangements.

For a smooth projective variety $X$ of dimension $n$ in a projective space ${\bb P}^N$ defined over an algebraically closed field $k$, the Gauss map is a morphism from $X$ to the Grassmannian of $n$-plans in ${\bb P}^N$sending $x \in X$ to the embedded tangent space $T_xX \subset {\bb P}^N$. The purpose of this paper is to prove the generic injectivity of Gauss maps in positive characteristic for two cases; (1) weighted complete intersections of dimension $n \geqslant 3$ of general type; (2) surfaces or 3-folds with $\mu$-semistable tangent bundles; based on a criterion of Kaji by looking at the stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman-Piene is true in case $X$ is of general type of dimension $n \geqslant 3$. The second result is a generalization of the injectivity for curves.