## Final Activity Report Summary - XTATADUCROHET (On some open problems about curves in Algebraic geometry)

During the three years of the project, two of which in the famous Tata Institute of Fundamental Research (TIFR) in Mumbai (India), I managed to acquire a wide and global picture of my research field which is algebraic geometry in positive characteristic and more specifically problems related to vector bundles over curves and moduli spaces. As far as my personal production is concerned, I focused on two topics.

The first dealt with the geometry of the rational map V induced by Frobenius action on rank 2 semi-stable vector bundles with trivial determinant over a general genus 2 curve X in odd characteristic p>0 at the level of moduli space M_X(2) of such bundles. This moduli space identifies with the 3-dimensional projective space and the semi-stable boundary identifies with the Kummer surface. In a first work expending some results from my thesis, I computed explicit equations for this map for p=3,5,7 and derived analogous results for genus 3 curves. To do so, I used Prym varieties associated to double étale covers. In a second work, I gave a characteristic free proof of the fact that the set H of stable bundle in the moduli space that are strictly semi-stable after pull back by Frobenius is an irreducible degree 2(p-1) divisor in M_X(2).

The second dealt with applications of Lange-and Stuhler's characterization of stable vector bundles over a projective variety X in positive characteristic as periodic vector bundles under pull-back by the absolute Frobenius of X. With I. Biswas, we extended this result to stable principal G-bundles for any connected algebraic group G defined on the prime subfield. With V. Mehta, we used a powerful result of E. Hrushovski to prove an anabelian analogue of the density of prime-to-p torsion points in the jacobian of a curve in characteristic p. Namely, the set of stable vector bundles of given rank and degree 0 over a curve X arising from a representation of the algebraic fundamental group form a Zariski dense subset in the corresponding moduli space.

The first dealt with the geometry of the rational map V induced by Frobenius action on rank 2 semi-stable vector bundles with trivial determinant over a general genus 2 curve X in odd characteristic p>0 at the level of moduli space M_X(2) of such bundles. This moduli space identifies with the 3-dimensional projective space and the semi-stable boundary identifies with the Kummer surface. In a first work expending some results from my thesis, I computed explicit equations for this map for p=3,5,7 and derived analogous results for genus 3 curves. To do so, I used Prym varieties associated to double étale covers. In a second work, I gave a characteristic free proof of the fact that the set H of stable bundle in the moduli space that are strictly semi-stable after pull back by Frobenius is an irreducible degree 2(p-1) divisor in M_X(2).

The second dealt with applications of Lange-and Stuhler's characterization of stable vector bundles over a projective variety X in positive characteristic as periodic vector bundles under pull-back by the absolute Frobenius of X. With I. Biswas, we extended this result to stable principal G-bundles for any connected algebraic group G defined on the prime subfield. With V. Mehta, we used a powerful result of E. Hrushovski to prove an anabelian analogue of the density of prime-to-p torsion points in the jacobian of a curve in characteristic p. Namely, the set of stable vector bundles of given rank and degree 0 over a curve X arising from a representation of the algebraic fundamental group form a Zariski dense subset in the corresponding moduli space.