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Let C$C$ be an algebraic curve of genus g$g$ and L$L$ a line bundle over C$C$. Let MSC(n,L)$\mathcal {M}S_C(n,L)$ and MOC(n,L)$\mathcal {M}O_C(n,L)$ be the moduli spaces of L$L$‐valued symplectic and orthogonal bundles, respectively, over C$C$ of rank n$n$. We construct rational curves of Hecke type on these moduli spaces which generalize the Hecke curves on the moduli space of vector bundles. As a main result, we show that these curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its by‐products, we show the non‐abelian Torelli theorem and compute the automorphism group of the moduli spaces. [ABSTRACT FROM AUTHOR]

We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V 5 and V 22 up to degree 3. We use torus localization for GW invariants computation, and use classical results for Hilbert schemes on V 5 and V 22 for DT invariants computation. From these computations, one can check correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by Cao-Maulik-Toda in genus 0. Also we can compute genus 1 GV invariants via the conjecture of Cao-Toda, which turned out to be 0. These fit into the fact that there are no smooth elliptic curves in V 5 and V 22 up to degree 3. [ABSTRACT FROM AUTHOR]

We study locally Cohen–Macaulay curves of low degree in the Segre threefold ℙ 1 × ℙ 1 × ℙ 1 {\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}} and investigate the irreducible and connected components, respectively, of the Hilbert scheme of them. We also apply the similar argument to the Segre threefold ℙ 2 × ℙ 1 {\mathbb{P}^{2}\times\mathbb{P}^{1}}. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the moduli space of stable pairs (or sheaves) on del Pezzo surface S is isomorphic to the Maruyama type transformation of a projective bundle of a quiver representation space. [ABSTRACT FROM AUTHOR]

Let be a smooth projective curve with genus . Let be the moduli space of stable rank two vector bundles on with a fixed determinant for . In this paper, as a generalization of Kiem and Castravet's works, we study the stable maps in with genus and degree . Let be a natural closed subvariety of which parametrizes stable vector bundles with a fixed subbundle for a line bundle on . We describe the stable map space . It turns out that the space consists of two irreducible components. One of them parameterizes smooth rational cubic curves and the other parameterizes the union of line and smooth conics. [ABSTRACT FROM AUTHOR]

Let M α ( d , χ ) be the moduli space of α -stable pairs ( s , F ) on the projective plane P 2 with Hilbert polynomial χ ( F ( m ) ) = d m + χ . For sufficiently large α (denoted by ∞ ), it is well known that the moduli space is isomorphic to the relative Hilbert scheme of points over the universal degree d plane curve. For the general ( d , χ ) , the relative Hilbert scheme does not have a bundle structure over the Hilbert scheme of points. In this paper, as the first non trivial such a case, we study the wall crossing of the α -stable pairs space when ( d , χ ) = ( 5 , 2 ) . As a direct corollary, by combining with Bridgeland wall crossing of the moduli space of stable sheaves, we compute the virtual Poincaré polynomial of M ∞ ( 5 , 2 ) . [ABSTRACT FROM AUTHOR]

We find sharp bounds on h0(F) for one-dimensional semistable sheaves F on a projective variety X. When X is the projective plane ℙ2, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a closed subset of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space. [ABSTRACT FROM AUTHOR]

Let M n be the Simpson compactification of twisted ideal sheaves I L , Q (1) where Q is a rank 4 quadratic hypersurface in P n and L is a linear subspace of dimension n − 2. This paper calculates the intersection Poincaré polynomial of M n using Kirwan's desingularization method. We obtain the intersection Poincaré polynomial of the moduli space for one-dimensional sheaves on del Pezzo surfaces of degree ≥ 8 by considering wall-crossings of stable pairs and complexes. [ABSTRACT FROM AUTHOR]