Belmans, Oberdieck, and Rennemo asked whether all unnatural automorphisms of Hilbert schemes of points on surfaces, i.e. those automorphisms which do not arise from the underlying surface, can be characterized by the fact that they do not preserve the diagonal of non-reduced subschemes. Sasaki recently published examples, independently discovered by the author, of automorphisms on the Hilbert scheme of two points of certain abelian surfaces which preserve the diagonal but are nevertheless unnatural, giving a negative answer to the question.

We construct additional examples of unnatural automorphisms for abelian surfaces which preserve the diagonal for the Hilbert scheme of an arbitrary number of points. The underlying abelian surfaces in these examples have Picard rank at least 2, and hence are not generic.

We prove the converse statement that all automorphisms are natural on the Hilbert scheme of two points for a principally polarized abelian surface of Picard rank 1. Additionally, we prove the same if the polarization has self-intersection a perfect square

Quot schemes are fundamental objects in the moduli theory of algebraic geometry. Quot schemes of surfaces admit natural perfect obstruction theories if we consider 1-dimensional quotients of trivial vector bundles. We study various virtual invariants of such Quot schemes using the structure of Seiberg-Witten invariants and Hilbert schemes of points.

The main result expresses the virtual Quot scheme invariants universally in terms of Seiberg-Witten invariants and certain cohomological data of a surface. When a curve class is of Seiberg-Witten length N, we prove the multiplicative structural formula for the generating series of equivariant virtual Quot scheme invariants in terms of the universal series and Seiberg-Witten invariants. Furthermore, the universal series are completely determined up to the change of variables. As an application, we prove the rationality of the homological and K-theoretic descendent series for any curve classes of Seiberg-Witten length N. Explicit formulas are available in several cases for specializations to the generating series of virtual Euler characteristics and virtual $\chi_{-y}$-genera.

For K3 surfaces, the usual virtual Quot scheme invariants vanish. We thus define and study the reduced invariants of Quot schemes. Rather surprisingly, we show that the reduced $\chi_{-y}$-genera of Quot schemes and Pair spaces are equal when N=1. This implies that the reduced $\chi_{-y}$-genera of Quot schemes are given by the Kawai-Yoshioka formula.

We study the virtual Segre and Verlinde series of Quot schemes as a variation of the non-virtual Segre and Verlinde series of Hilbert schemes of points. Analogously to the conjectural algebraicity of the Segre and Verlinde series of Hilbert schemes of points, we prove that the virtual Segre and Verlinde series of Quot schemes are algebraic for any curve classes of Seiberg-Witten length N. Furthermore, we prove the virtual Segre and Verlinde correspondence relating the universal series. This, in particular, matches the virtual Segre and Verlinde series for punctual Quot schemes up to a sign.

I present formulas for the Euler characteristics of tautological sheaves over the punctual Quot scheme, which parameterizes zero-dimensional quotients of a fixed vector bundle over curves. We observe a striking similarity with the formulas for the Hilbert scheme of points on surfaces. Furthermore, we study the Quot schemes of higher rank quotients for a genus-zero curve. We calculate the holomorphic Euler characteristics of Schur bundles and tautological bundles over Quot schemes. These formulas can be considered a generalization of the formulas for Grassmannians, which were obtained using the Borel-Weil-Bott theorem. Additionally, we show non-trivial vanishing results using these formulas.

The symplectic (or orthogonal) Grassmannian parameterizes isotropic subspaces of a vector space endowed with a symplectic (or symmetric) bilinear form. I study the intersection theory of the symplectic and orthogonal isotropic Quot schemes. In particular, I construct a virtual fundamental class for these Quot schemes and find explicit formulas for certain intersection numbers. I also calculate the Gromov-Ruan-Witten invariants of the corresponding Grassmannians and compare the answers with those for the isotropic Quot schemes.

Moduli spaces of K3 surfaces are fundamental objects in algebraic geometry. Elliptic K3 surfaces are K3 surfaces with elliptic fibration structure, and they are of particular interest due to their rich geometry. The moduli space of elliptic K3 surfaces can be studied using the theory of Weierstrass models. In this dissertation, we study the topology and intersection theory of the moduli space of elliptic K3 surfaces.

We compute the Poincar e polynomial of the moduli space of elliptic K3 surfaces. The main idea is constructing a compactification using the Weierstrass models, this compactification is a GIT quotient. We adapt Kirwan's blowup machinery to weighted projective space to compute the Poincar e polynomial. We find the cohomology is mostly concentrated in the even degrees, but there is one odd degree class in degree $33$.

We also study the Chow ring of the moduli space of elliptic surfaces of degree $N\geq 2$. We conclude that the Chow ring of the moduli space of elliptic surfaces is always generated by two classes. Furthermore, explicit relations between these classes are given, the Poincar e polynomial for the Chow ring is the same for any $N\geq 2$ and the ring is Gorenstein with socle in degree $16$. When $N=2$, we obtain the Chow ring for the moduli space of elliptic K3 surfaces, we conclude that the Chow ring in this case is tautological.

Finally, we present localization computations on the relative Quot scheme over the moduli space of elliptic K3 surfaces. Our calculations are sufficient to determine the divisorial $\kappa$-classes in terms of the Hodge class. We also represent one Noether-Lefschetz divisor in terms of the Hodge class, which agrees with the modularity nature of the Noether-Lefschetz divisors.

There are three parts in this thesis. First, we generalize the class of tropical

curves from trivalent to 3-colorable which can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by $\Gamma$.

Second, we prove the equality of two canonical bases of a rank 2 cluster

algebra, the greedy basis of Lee-Li-Zelevinsky and the theta basis of Gross-Hacking-Keel-Kontsevich.

Third, we link up scattering diagrams D with quiver representations of

corresponding quivers Q. We define a notion of good crossing of broken lines $\gamma$ on D. Then we show if $\gamma$ has good crossing over D, then it goes in the opposite direction of the Auslander-Reiten quiver of Q. Then we give a stratification of quiver representations by the bendinga of $\gamma$.

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We compute the Picard group of the moduli stack of genus zero stable quasimaps

to projective space, Grassmannians, and any

flag variety in the case of more than 2

markings. Furthermore, in the case of exactly 2 markings, we calculate the Picard

group of the moduli stack of genus zero stable quasimaps to projective space, Grassmannians,

and to partial

flag varieties where the ranks of the subspaces differ by

more than 1. The first two moduli stacks mentioned are the moduli stacks of stable

quotients, constructed by Alina Marian, Dragos Oprea, and Rahul Pandharipande.

The latter is a generalization of this theory, due to Ionut-Ciocan Fontanine, Bumsig

Kim, and Davesh Maulik. Projectivity of the coarse moduli space is proved first.

The Picard rank is obtained using a torus action on the moduli stack to perform

tangent space calculations. When the number of markings is greater than or equal to 3, generators are

determined by a geometric analysis of the interior of the moduli stack. When the

number of markings is 2, generators and relations are found by intersecting with

curves.