Narrow Results

In this paper we construct an abelian fibration over P³ on the Hilbert cube of the primitive K3 surface of genus 9. After the abelian fibration constructed by Hassett and Tschinkel on the Hilbert square on the primitive K3 surface of genus 5, this is the second example where the abelian fibration is constructed directly on . The recent more general result of Sawon proves the existence of an abelian fibration on the Hilbert scheme of a primitive K3 surface S of degree 2g - 2 = m²(2n - 2). Our example provides an alternative proof in the case m = 2, n = 3. Furthermore we identify the general fiber with the Hilbert scheme of twisted cubic curves in a Fano 3-fold of genus 9, and interpret the addition law on this variety.

In this paper, we study the Gromov–Witten theory of the Hilbert schemes X^[n] of points on a smooth projective surface X with positive geometric genus p_g. For fixed distinct points x₁, …, x_n-1 ∈ X, let β_n be the homology class of the curve {ξ + x₂ + ⋯ + x_n-1 ∈ X^[n] | Supp(ξ) = {x₁}}, and let β_KX be the homology class of {x + x₁ + ⋯ + x_n-1 ∈ X^[n] | x ∈ K_X}. Using cosection localization technique due to Y. Kiem and J. Li, we prove that if X is a simply connected surface admitting a holomorphic differential two-form with irreducible zero divisor, then all the Gromov–Witten invariants of X^[n] defined via the moduli space of stable maps vanish except possibly when β is a linear combination of β_n and β_KX. When n = 2, the exceptional cases can be further reduced to the Gromov–Witten invariants: with and d ≤ 3, and with d ≥ 1. When , we show that which is consistent with a well-known formula of C. Taubes. In addition, for an arbitrary surface X and d ≥ 1, we verify that .

In this note we prove that the Fourier–Mukai transform ${\mathrm{\Phi }}_{{𝒰}}$ of the universal family of the moduli space ${{ℳ}}_{{\mathbb{P}}^2}(4,1,3)$ is not fully faithful.

In this paper, we study the Gromov–Witten theory of the Hilbert scheme $X^{[2]}$ of two points on an elliptic surface $X$. Assume that $\left|K_X\right|$ contains an element supported on the smooth fibers of $X$. By analyzing the degeneracy locus and localized virtual cycle arising from the cosection localization theory of Kiem and Li [Y. Kiem and J. Li, Gromov–Witten invariants of varieties with holomorphic 2-forms, preprint; Y. Kiem and J. Li, Localizing virtual cycles by cosections, J. Amer. Math. Soc. 26 (2013) 1025–1050], we determine the $1$-point genus-$0$ Gromov–Witten invariant ${〈w〉}_{0,d({\beta }_f-2{\beta }_2)}^{X[2]}$ up to some rational number $m(d,X)$ depending only on $d$ and $X$, where $w\in H^4(X^{[2]},ℂ)$, $d\ge 1$, $f$ is a smooth fiber of $X$, ${\beta }_f=x_0+f\subset X^{[2]}$ with $x_0\in X-f$ being a fixed point, and ${\beta }_2=\{\xi \in X^{[2]}|\mathrm{\text{Supp}}(\xi )=\{x_0\}\}$. Moreover, we propose a conjecture regarding $m(d,X)$, and prove that the conjecture is true for $X=C\times E$ where $E$ is an elliptic curve and $C$ is a smooth curve.

We study the Hilbert scheme of points on a cusp curve, mostly the punctual Hilbert scheme which parameterizes m points supported at a cusp. We show that the reduced punctual scheme is isomorphic to ℙ¹ and the Hilbert scheme has one singularity along .

The Hilbert scheme ${\mathrm{\text{Hilb}}}^n{ℂ}^3$ of n points on ${ℂ}^3$ can be expressed as the critical locus of a regular function on a smooth variety ${𝒳}$. Recent development in birational geometry suggests a study of singularities of the pair $({𝒳},{\mathrm{\text{Hilb}}}^n{ℂ}^3)$ using jet schemes. In this paper, we use a comparison between ${\mathrm{\text{Hilb}}}^n{ℂ}^3$ and the scheme $C_{3,n}$ of three commuting $n\times n$ matrices to estimate the log canonical threshold of $({𝒳},{\mathrm{\text{Hilb}}}^n{ℂ}^3)$. As a consequence, we see that although both $dim{𝒳}$ and $dim{\mathrm{\text{Hilb}}}^n{ℂ}^3$ have asymptotic growth $O(n^2)$, the largest multiplicity of any points on ${\mathrm{\text{Hilb}}}^n{ℂ}^3$ has at most linear growth $O(n)$.

For any integer d ≥ 5, the Noether–Lefschetz locus, denoted NL_d, parametrizes smooth degree d surfaces in ℙ³ with Picard number at least 2. It is well-known (due to works of Voisin, Green and others) that the largest irreducible component of NL_d is of codimension (in the space of all smooth surfaces in ℙ³ of degree d) equal to d-3 and parametrizes surfaces containing a line. In this article we study for an integer 3 ≤ r < d, the sub-locus of NL_d, denoted NL_r,d, parametrizing surfaces with Picard number at least r. A conjecture of Griffiths and Harris states the largest component of NL_r,d is of codimension and the irreducible component of NL_r,d parametrizing the surfaces containing r - 1 coplanar lines is of this codimension. We prove this statement in the case r ≪ d.

The Noether–Lefschetz theorem asserts that any curve in a very general surface $X$ in ${\mathbb{P}}^3$ of degree $d\ge 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that under some conditions, which replace the condition $d\ge 4$, a very general surface in a simplicial toric threefold ${\mathbb{P}}_{\mathrm{\Sigma }}$ (with orbifold singularities) has the same Picard number as ${\mathbb{P}}_{\mathrm{\Sigma }}$. Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in ${\mathbb{P}}_{\mathrm{\Sigma }}$ in a linear system of a Cartier ample divisor with respect to a $(-1)$-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether–Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense.

Let $\rho :\mathbb{Z}/k\mathbb{Z}\to SL(2,ℂ)$ be a representation of a finite abelian group and let ${\mathrm{\Theta }}^{\text{gen}}\subset {\mathrm{\text{Hom}}}_{\mathbb{Z}}(R(\mathbb{Z}/k\mathbb{Z}),\mathbb{Q})$ be the space of generic stability conditions on the set of $G$-constellations. We provide a combinatorial description of all the chambers $C\subset {\mathrm{\Theta }}^{\mathrm{\text{gen}}}$ and prove that there are $k!$ of them. Moreover, we introduce the notion of simple chamber and we show that, in order to know all toric $G$-constellations, it is enough to build all simple chambers. We also prove that there are $k\cdot 2^{k-2}$ simple chambers. Finally, we provide an explicit formula for the tautological bundles ${{ℛ}}_C$ over the moduli spaces ${{ℳ}}_C$ for all chambers $C\subset {\mathrm{\Theta }}^{\mathrm{\text{gen}}}$ which only depends upon the chamber stair which is a combinatorial object attached to the chamber $C$.