Hilbert scheme of points on a surface

In these lectures, we are interested in the case of dimX = 2, and in the following we assume that X is nonsingular and dimX = 2 whenever otherwise stated. In this case, Xl has especially nice properties as the next theorem shows. 

Theorem 1.8 (Fogarty [17]). Suppose X is nonsingular and dimX = 2, then the following holds. 

To prove the smoothness of it is enough to show that the dimension of does 

(1) Let Z G and Jz the corresponding ideal. The Zariski tangent space of X[ 1 at Z is given by 

To prove the smoothness of X n it is enough to show that the dimension of TZX does not depend on Z G From the exact sequence 

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In this chapter, we collect basic facts on the Hilbert scheme of points on a surface. We do not assume the field fc is C unless mentioned. 

The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), 193-207. 

Question 4.6 (Hitchin). Gonsider a hnite group action on a K3 surface which preserves a hyper-Kahler structure. (Such actions were classified by Mukai [58].) It naturally induces the action on the Hilbert scheme of points on the K3 surface. Its fixed point component is a compact hyper-Kahler manifold as in 4.2. Is the component a new hyper-Kahler manifold The known compact irreducible hyper-Kahler manifolds are equivalent to the Hilbert scheme of points on a /F3 surface, or the higher order Kummar variety (denoted by Kr in [6]) modulo deformation and birational modification, (cf. [57, p.l68j)... 

The Hilbert scheme of points on the cotangent bundle of a Riemann surface has a natural holomorphic symplectic structure together with a natural C -action. In this case, the unstable manifold is very important since it becomes a Lagrangian submanifold. The same kind of situation appears in many cases, for example when one studies the moduli space of Higgs bundle or the quiver varieties [62], and it is worth explaining this point before studying the specific example. 

Now we shall study the Hilbert scheme of points on the cotangent bundle of a Riemann surface. Let E be a Riemann surface and T E its cotangent bundle. There exists a natural holomorphic symplectic form uc on T E. The multiplication by a complex number on each fiber gives a natural C -action on T E, and with respect to this action we have "(p uJc = tuc for t E C, where we denote the action of t by T E T E. As explained in Theorem 1.10, the Hilbert scheme inherits a holomorphic symplectic form and... 

In the spring of 1996, I gave a series of lectures on the Hilbert schemes of points on surfaces at Department of Mathematical Sciences, University of Tokyo. 

First, we recall the definition of the Hilbert scheme in general (not necessarily of points, nor on a surface). Let X be a projective scheme over an algebraically closed field k and Gx(l) an ample line bundle on X. We consider the contra,variant functor Hilbx from the... 

In this chapter, we shall prove the following formula for Poincare polynomial of Hilbert scheme of n-points on a quasi-projective nonsingular surface X ... 

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