Let n≥4, 2≤r≤n−2 and e≥1. We show that the intersection of the locus of degree e morphisms from P1 to G(r,n) with the restricted universal sub-bundles having a given splitting type and the locus of ...degree e morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse along at least one component of the intersection. As a consequence, we get that the locus of degree e morphisms from P1 to G(r,n) with the restricted tangent bundle having a given splitting type need not always be irreducible.
Abstract
Let $r \geq 2$ be an integer, and let $a$ be an integer coprime to $r$. We show that if $c_2 \geq n + \left \lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right \rfloor $, then the $2n$th ...Betti numbers of the moduli space $M_{\mathbb{P}^2,H}(r,aH,c_2)$ stabilize where $H = c_1(\mathcal{O}_{\mathbb{P}^2}(1))$.
Non-globally Generated Bundles on Curves Kopper, John; Mandal, Sayanta
International mathematics research notices,
01/2023, Volume:
2023, Issue:
2
Journal Article
Peer reviewed
Open access
Abstract
We describe the locus of stable bundles on a smooth genus $g$ curve that fail to be globally generated. For each rank $r$ and degree $d$ with $rg<d<r(2g-1)$, we exhibit a component of the ...expected dimension. We show, moreover, that no component has larger dimension and give an explicit description of those families of smaller dimension than expected. For large-enough degrees, we show that the locus is irreducible.
This thesis is based on work done on two different problems. The first problem is regarding restricted tangent bundles of the Grassmannian to rational curves. Let n ≥ 4, 2 <= r <= n-2 and e ≥ 1. We ...show that the intersection of the locus of degree e morphisms from P^1 to G(r, n) with the restricted universal sub-bundles having a given splitting type and the locus of degree e morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse. As a consequence, we get that the locus of degree e morphisms from P^1 to G(r, n) with the restricted tangent bundle having a given splitting type need not always be irreducible.The second problem is regarding the Betti numbers of the moduli space of sheaves on the projective plane. Let r >= 2 be an integer, and let a be an integer coprime to r. We show that if c2 ≥ n+ (r-1/2r)a2 + 1/2(r2 + 1), then the 2nth Betti number of the moduli space MP2, oP2 (1)(r,OP2(a), c2) stabilizes.
Let $n\geq 4$, $2 \leq r \leq n-2$ and $e \geq 1$. We show that the
intersection of the locus of degree $e$ morphisms from $\mathbb{P}^1$ to
$G(r,n)$ with the restricted universal sub-bundles having ...a given splitting
type and the locus of degree $e$ morphisms with the restricted universal
quotient-bundle having a given splitting type is non-empty and generically
transverse. As a consequence, we get that the locus of degree $e$ morphisms
from $\mathbb{P}^1$ to $G(r,n)$ with the restricted tangent bundle having a
given splitting type need not always be irreducible.
Let $r \geq 2$ be an integer, and let $a$ be an integer coprime to $r$. We
show that if $c_2 \geq n + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 +
1) \right\rfloor$, then the $2n$th Betti ...number of the moduli space
$M_{\mathbb{P}^2,H}(r,aH,c_2)$ stabilizes, where $H =
c_1(\mathcal{O}_{\mathbb{P}^2}(1))$.
We describe the locus of stable bundles on a smooth genus \(g\) curve that fail to be globally generated. For each rank \(r\) and degree \(d\) with \(rg<d<r(2g-1)\), we exhibit a component of the ...expected dimension. We show moreover that no component has larger dimension and give an explicit description of those families of smaller dimension than expected. For large enough degrees, we show that the locus is irreducible.
Let \(r \geq 2\) be an integer, and let \(a\) be an integer coprime to \(r\). We show that if \(c_2 \geq n + \left\lfloor \frac{r-1}{2r}a^2 + \frac{1}{2}(r^2 + 1) \right\rfloor\), then the \(2n\)th ...Betti number of the moduli space \(M_{\mathbb{P}^2,H}(r,aH,c_2)\) stabilizes, where \(H = c_1(\mathcal{O}_{\mathbb{P}^2}(1))\).
Let \(n\geq 4\), \(2 \leq r \leq n-2\) and \(e \geq 1\). We show that the intersection of the locus of degree \(e\) morphisms from \(\mathbb{P}^1\) to \(G(r,n)\) with the restricted universal ...sub-bundles having a given splitting type and the locus of degree \(e\) morphisms with the restricted universal quotient-bundle having a given splitting type is non-empty and generically transverse. As a consequence, we get that the locus of degree \(e\) morphisms from \(\mathbb{P}^1\) to \(G(r,n)\) with the restricted tangent bundle having a given splitting type need not always be irreducible.
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