Stability of Vector Bundles and Rational Curves

This thesis is based on two research papers, each addressing distinct but interconnected topics within algebraic geometry. In the first part, we study the stability of general kernel bundles on projective space P^n. Let a, b, d > 0 be integers. A kernel bundle E_{a,b} on P^n is defined as the kernel of a surjective map φ from O_{P^n}(-d)^a to O_{P^n}^b. Here, φ is represented by a b×a matrix (f_{ij}), where the entries f_{ij} are polynomials of degree d. We give sufficient conditions for semistability of a general kernel bundle on P^n, in terms of its Chern class. In the second part, we study whether a given morphism f from the tangent bundle of P^1 to a balanced vector bundle of degree (n+1)d is induced by the restriction of the tangent bundle T_{P^n} to a rational curve of degree d in P^n. We propose a conjecture on this problem based on Mathematica computations of some examples and provide computer-assisted proof of the conjecture for certain values of n and d.