### A fixed point formula for varieties over finite fields.

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Let $X$ be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of $X$ is contained in at most one rational curve of minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.

We show that if the degree of a nonsingular projective variety is high enough, maximization of any of the most important numerical invariants, such as class, Betti number, and any of the Chern or middle Hodge numbers, leads to the same class of extremal varieties. Moreover, asymptotically (say, for varieties whose total Betti number is big enough) the ratio of any two of these invariants tends to a well-defined constant.

The existence of common zero of a family of polynomials has led to the study of inertial forms, whose homogeneous part of degree 0 constitutes the ideal resultant. The Kozsul and Cech cohomologies groups play a fundamental role in this study. An analogueous of Hurwitz theorem is given, and also, one finds a N. H. McCoy theorem in a particular case of this study.

Using the notion of uniruledness we indicate a class of algebraic varieties which have a stronger version of the cancellation property. Moreover, we give an affirmative solution to the stable equivalence problem for non-uniruled hypersurfaces.

Given a covering family $V$ of effective 1-cycles on a complex projective variety $X$, we find conditions allowing one to construct a geometric quotient $q:X\to Y$, with $q$ regular on the whole of $X$, such that every fiber of $q$ is an equivalence class for the equivalence relation naturally defined by $V$. Among other results, we show that on a normal and $\mathbb{Q}$-factorial projective variety $X$ with canonical singularities and $dimX\le 4$, every covering and quasi-unsplit family $V$ of rational curves generates a geometric extremal...