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We generalize the classical semi-continuity theorem for GIT (semi)stable loci under variations of linearizations to a relative situation of an equivariant projective morphism from X to an affine base S. As an application to moduli problems, we consider degenerations of Hilbert schemes, and give a conceptual interpretation of the (semi)stable loci of the degeneration families constructed by Gulbrandsen-Halle-Hulek.
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In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert scheme as subscheme of a suitable Grassmannian and in Chapter 5 we determine a new set of equations of degree lower than the degree of equations known so far. In Chapter 2 we study the most important objects used to project algorithmic techniques, namely Borel-fixed ideals. We determine an algorithm computing all the saturated Borel-fixed ideals with Hilbert polynomial assigned and we investigate their combinatorial properties. In Chapter 3 we show a new type of flat deformations of Borel-fixed ideals which lead us to give a new proof of the connectedness of the Hilbert scheme. In Chapter 4 we construct families of ideals that generalize the notion of family of ideals sharing the same initial ideal with respect to a fixed term ordering. Some of these families correspond to open subsets of the Hilbert scheme and can be used to a local study of the Hilbert scheme. In Chapter 6 we deal with the problem of the connectedness of the Hilbert scheme of locally Cohen-Macaulay curves in the projective 3-space. We show that one of the Hilbert scheme considered a good candidate to be non-connected, is instead connected. Moreover there are three appendices that present and explain how to use the implementations of the algorithms proposed.
We take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety $X$ as a projective completion of the non-reductive quotient of holomorphic map germs from the complex line into $X$ by polynomial reparametrisations. Using an algebraic model of this quotient coming from global singularity theory we develop an iterated residue formula for tautological integrals over curvilinear Hilbert schemes.
We express nested Hilbert schemes of points and curves on a smooth projective surface as virtual resolutions of degeneracy loci of maps of vector bundles on smooth ambient spaces. We show how to modify the resulting obstruction theories to produce the virtual cycles of Vafa-Witten theory and other sheaf-counting problems. The result is an effective way of calculating invariants (VW, SW, local PT and local DT) via Thom-Porteous-like Chern class formulae.
We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain positivity conditions, we relate the DT invariants to Carlsson-Okounkov formulas for the twisted Eulers number of the punctual Hilbert schemes of nonsingular surfaces, and conclude they have a modular property.
In this paper we study the tangent spaces of the smooth nested Hilbert scheme $ Hil{n,n-1}$ of points in the plane, and give a general formula for computing the Euler characteristic of a $TT^2$-equivariant locally free sheaf on $Hil{n,n-1}$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients . We call this conjecturally positive polynomial as textsl{the nested $q,t$-Cat alan series}, for it has many conjectural properties similar to that of the $q,t $-Catalan series.
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