Derived categories of coherent sheaves and motives of K3 surfaces

Let X and Y be smooth complex projective varieties. Orlov conjectured that if X and Y are derived equivalent then their motives M(X) and M(Y) are isomorphic in Voevodsky's triangulated category of geometrical motives with rational coefficients. In this paper we prove the conjecture in the case X is a K3 surface admitting an elliptic fibration (a case that always occurs if the Picard rank of X is at least 5) with finite-dimensional Chow motive. We also relate this result with a conjecture by Huybrechts showing that, for a K3 surface with a symplectic involution, the finite-dimensionality of its motive implies that the involution acts as the identity on the Chow group of 0-cycles. We give examples of pairs of K3 surfaces with the same finite-dimensional motive but not derived equivalent.Comment: 18 page

On some aspects of polynomial dynamical systems

The aim of this work is to study exact algebraic criteria local/global observability ([HK77], [Ino77]) for polynomial dynamical system by means of algebraic geometry and computational commutative algebra in the vein of [SR76], [Son79a], [Son79b], [Bai80], [Bai81], [Bar95], [Bar99], [Nes98], [Tib04], [KO13], [Bar16]. A key point in this topic is to work with polynomials with real coefficients and their real roots instead of their complex roots, as it is usually the case ([CLO15], [KR00]). A central concept is then the real radical of an ideal [BN93], [Neu98], [LLM+13], along with the Krivine- Dubois-Risler real nullstellensatz for polynomial rings [Kri64], [Dub70], [Ris70], [BCR98]. Underestimating this point leads to incorrect results (see, e.g. [Bar16] remark on [KO13]). This thesis is therefore devoted to set the necessary algebraic tools in the right context and level of generality (i.e. real algebra and real algebraic geometry) for applications to our dynamical systems and to further develop their exploit in this context. The first two chapters set the algebraic and algebraic geometry preliminaries. The third chapter is devoted to the applications of the previous algebraic concepts to the study of the ob- servability of polynomial dynamical systems. In the last chapter an approach to the construction of Lyapunov funtions to prove stability in estimation problems is presented

On a theorem of Faltings on formal functions

In 1980, Faltings proved, by deep local algebra methods, a local result regarding formal functions which has the following global geometric fact as a consequence. Theorem: Let k be an algebraically closed field (of any characteristic). Let Y be a closed subvariety of a projective irreducible variety X defined over k. Assume that X \subseteq P^n, dim(X)=d>2 and Y is the intersection of X with r hyperplanes of P^n, with r \le d-1. Then, every formal rational function on X along Y can be (uniquely) extended to a rational function on X. Due to its importance, the aim of this paper is to provide two elementary global geometric proofs of this theorem.Comment: 9 page

On the minimum distance of AG codes, on Weierstrass semigroups and the smoothability of certain monomial curves in 4-Space

In this paper we treat several topics regarding numerical Weierstrass semigroups and the theory of Algebraic Geometric Codes associated to a pair $(X, P)$, where $X$ is a projective curve defined over the algebraic closure of the finite field $F_q$ and P is a $F_q$-rational point of $X$. First we show how to evaluate the Feng-Rao Order Bound, which is a good estimation for the minimum distance of such codes. This bound is related to the classical Weierstrass semigroup of the curve $X$ at $P$. Further we focus our attention on the question to recognize the Weierstrass semigroups over fields of characteristic 0. After surveying the main tools (deformations and smoothability of monomial curves) we prove that the semigroups of embedding dimension four generated by an arithmetic sequence are Weierstrass.Comment: 30 pages, presented at CAAG 2010 (Bangalore, India

Schur finiteness and nilpotency

Let A be a Q-linear pseudo-abelian rigid tensor category. A notion of finiteness due to Kimura and (independently) O'Sullivan guarantees that the ideal of numerically trivial endomorphism of an object is nilpotent. We generalize this result to special Schur-finite objects. In particular, in the category of Chow motives, if X is a smooth projective variety which satisfies the homological sign conjecture, then Kimura-finiteness, a special Schur-finiteness, and the nilpotency of CH^{ni}(X^i\times X^i)_{num} for all i (where n=dim X) are all equivalent