Doctor of Philosophy

dissertationThe main object of study in this thesis is the Grothendieck Quot scheme. LetXbe aprojective variety overC, letVbe a coherent sheaf onX, and letρbe a cohomology classonX. The Quot scheme Quot(V,ρ)is a projective scheme that parametrizes coherent sheafquotientsV#16;FwhereFhas Chern characterρ. Choosingρand the Chern character ofVto satisfy a certain orthogonality condition, Quot(V,ρ)is expected to be a finite collectionof points. One can ask whether Quot(V,ρ)is indeed finite whenVis general in moduli. Ifso, then one can try to enumerate the points of Quot(V,ρ). These counts of points of finiteQuot schemes yield interesting formulas and can be used to study strange duality.WhenXis a curve, Marian and Oprea proved that generalVdo produce finite Quotschemes, whose points are counted by the Verlinde formula. We show that these enumer-ative invariants can be viewed as certain closed invariants inside a weighted topologicalquantum field theory (TQFT) that encodes the intersection numbers of Schubert varietieson all (not only finite) Quot schemes of general vector bundles on curves. This weightedTQFT contains both the small quantum cohomology of the Grassmannian and a TQFT ofWitten that is known to compute the Verlinde numbers.WhenXis a del Pezzo surface, even the existence of finite Quot schemes is not known.OnP2, we use exceptional resolutions of sheaves to prove that Quot(V,ρ)is finite whenρis the Chern character of an ideal sheaf of points, the orthogonality condition is satisfied,andVis general in moduli. On general del Pezzo surfaces, we use multiple point formulasto compute the expected number of points of Quot schemes that are expected to be finite,whereρis once again the Chern character of an ideal sheaf of points. The formulas agreewith a power series computing Euler characteristics of line bundles on Hilbert schemes ofpoints, thus providing evidence for strange duality on del Pezzo surfaces