arXiv subjects

Neelarnab Raha

Publications and source records attributed to Neelarnab Raha.

5 recordsLinked to original sources

arXiv2025

Algebraic hyperbolicity of subvarieties of homogeneous varieties

We study the algebraic hyperbolicity of certain subvarieties of homogeneous varieties, building on the techniques introduced by Coskun-Riedl, Yeong and Mioranci. This generalizes earlier known results for hypersurfaces to higher codimensions. In particular, we observe that if $X=X_1\cap\cdots\cap X_k$ is a very general complete intersection of degree $d_j$ hypersurfaces $X_j$ in $\mathbb{P}^n$ with $k\leq n-2$, then $X$ is algebraically hyperbolic if $\sum d_j\ge 2n-k$, and $X$ is not algebraically hyperbolic if $\sum d_j\le 2n-k-2$.

math.AG↗

arXiv2025

Higher rank Clifford's theorem on the smooth quadric

Brill-Noether theory of curves has played a crucial role in the study of curves and their moduli since the 19th century, and has been extensively studied by several authors. Clifford's theorem provides a starting point in determining the emptiness of Brill-Noether loci by providing an upper bound on $h^0(L)$ for a line bundle $L$ on a smooth curve $C$ in terms of the degree of $L$. It also characterizes the cases for which equality holds. In this paper, we prove an analogous result for higher rank sheaves on $\mathbb{P}^1\times\mathbb{P}^1$. Depending on how nice the first Chern class is, and whether the sheaf has global generation properties, we prove sharp upper bounds on $h^0(E)$ for slope semistable sheaves $E$ in terms of $\operatorname{rk}(E)$ and $c_1(E)$. We also find that any $E$ achieving the bound is a twist of a Steiner-like bundle, or closely related to such a bundle. As part of our investigation, we show that general extensions of stable vector bundles on elliptic curves and del Pezzo surfaces are semistable.

math.AG↗

arXiv2025

Nef cones of Hilbert schemes of points on some K3 surfaces

We illustrate the typical usage of Bayer and Macr\`{i}'s Positivity Lemma to compute the nef cones of the Hilbert schemes $X^{[n]}$ by combining the Bridgeland stability methods (for large $n$) and classical methods (for small $n$). We use Mori dream K3 surfaces $X$ of Picard rank $2$ as our working example. We also compute the nef cones of the nested Hilbert schemes $X^{[n,n+1]}$ for such $X$, for large $n$.

math.AG↗

arXiv2024

Brill-Noether theory on the projective plane for bundles with many sections

The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. Let $E$ be a semistable sheaf on the projective plane. In this paper, we give an upper bound for $h^0(E)$ in terms of the rank $r$ and the slope $\mu$ of $E$. We show that the bound is achieved precisely when $E$ is a twist of a Steiner bundle. We classify the sheaves $E$ such that $h^0(E)$ is sufficiently close to the upper bound. We determine the nonemptiness, irreducibility and dimension of the Brill-Noether loci in the moduli spaces of sheaves with $h^0(E)$ in this range. When they are nonempty, these Brill-Noether loci are irreducible though almost always of larger than the expected dimension.

math.AG↗

arXiv2023

Ample cones of Hilbert schemes of points on hypersurfaces in $\mathbb{P}^3$

Let $X$ be a very general degree $d\geq 5$ hypersurface in $\mathbb{P}^3$. We compute the ample cone of the Hilbert scheme $X^{[n]}$ of $n$ points on $X$ for various small values of $n$ (the answer is already known for large $n$). We obtain complete answers in some cases and find lower bounds in certain others. We also observe that in the case of $X^{[2]}$ for quintic hypersurfaces $X$, the existence (or absence) of hyperplane sections with points of high multiplicity also plays a role in the answer to the question at hand, in contrast with cases known earlier. Finally, in the case that a degree $d\geq3$ smooth hypersurface $X$ contains a line, we compute the nef cone of $X^{[n]}$ in a slice of the N\'eron-Severi space.

math.AG↗