arXiv subjects
Publications and source records attributed to Ananyo Dan.
Let $K$ be the fraction field of a Henselian discrete valuation ring with algebraically closed residue field $k$. In this article we give a sufficient criterion for a projective variety over such a field to have index $1$.
In this article, we study the Hilbert scheme of generically non-reduced curves in $\mathbb{P}^3$. We prove the existence of generically non-reduced curves in $\mathbb{P}^3$ for which there exist infinitesimal deformations of the curve that do not induce deformations of the associated reduced scheme. We show that such infinitesimal deformations contribute to the non-reducedness of the corresponding Hilbert scheme. We introduce the notion of extension of curves and prove that such infinitesimal deformations (such that the associated reduced scheme does not deform) are inherited by the extended curve. Finally, we give examples of extension of curves.
In this article, we study subloci of solvable curves in $\mathcal{M}_g$ which are contained in either a K3-surface or a quadric or a cubic surface. We give a bound on the dimension of such subloci. In the case of complete intersection genus $g$ curves in a cubic surface, we show that a general such curve is solvable.
Let $E_G$ be a holomorphic principal $G$-bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D \subset X$, and for each $x\in D$ fix $w_x \in \text{ad}(E_G)_x$. Let $T$ be a maximal torus in the group of all holomorphic automorphisms of $E_G$. We give a necessary and sufficient condition for the existence of a $T$-invariant logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$. We also give a necessary and sufficient condition for the existence of a logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$, under the assumption that each $w_x$ is $T$-rigid.
A theorem of Weil and Atiyah says that a holomorphic vector bundle $E$ on a compact Riemann surface $X$ admits a holomorphic connection if and only if the degree of every direct summand of $E$ is zero. Fix a finite subset $S$ of $X$, and fix an endomorphism $A(x) \in \text{End}(E_x)$ for every $x \in S$. It is natural to ask when there is a logarithmic connection on $E$ singular over $S$ with residue $A(x)$ at every $x \in S$. We give a necessary and sufficient condition for it under the assumption that the residues $A(x)$ are rigid.
We know that semi-regular sub-varieties satisfy the variational Hodge conjecture i.e., given a family of smooth projective varieties $π:\mathcal{X} \to B$, a special fiber $\mathcal{X}_o$ and a semi-regular subvariety $Z \subset \mathcal{X}_o$, the cohomology class corresponding to $Z$ remains a Hodge class (as $\mathcal{X}_o$ deforms along $B$) if and only if $Z$ remains an algebraic cycle. In this article, we investigate examples of such sub-varieties. In particular, we prove that any smooth projective variety $Z$ of dimension $n$ is a semi-regular sub-variety of a smooth projective hypersurface in $\mathbb{P}^{2n+1}$ of large enough degree.
Given a family $π:\mc{X} \rightarrow B$ of smooth projective varieties, a closed fiber $\mc{X}_o$ and an invertible sheaf $\mc{L}$ on $\mc{X}_o$, we compare the Hodge locus in $B$ corresponding to the Hodge class $c_1(\mc{L})$ with the locus of points $b\,\in\, B$ such that $\mc{L}$ deforms to an invertible sheaf $\mc{L}_b$ on $\mc{X}_b$ with at least $h^0(\mc{L})$--dimensional space of global sections (it is a Brill-Noether type locus associated to $\mc{L}$). We finally give an application by comparing the Brill-Noether locus to a family of curves on a surface passing through a fixed set of points.
For a fixed integer $d \ge 5$, the Noether-Lefschetz locus parametrizes smooth degree $d$ complex hypersurfaces in $\mathbb{P}^3$ with Picard number greater than $1$. There are infinitely many irreducible components of this locus. The aim of this article is to understand the scheme structure of these components. In particular, we study when is such a component non-reduced. We give a criterion for such a component to be non-reduced. Finally, we give examples of such components.
For a fixed integer $d$, we study here the locus of degree $d$ hypersurfaces $X$ in $\mathbb{P}^{2n+1}$ such that $H^{2n}(X,\mathbb{Q}) \cap H^{n,n}(X,\mathbb{C}) \not= \mathbb{Q}$. We call this locus \textit{the Noether-Lefschetz locus}. Any irreducible component of this locus is locally a Hodge locus. So, we see that the study of this locus is very closely related to the variational Hodge conjecture. In this article we show that the Hodge locus corresponding to the cohomology class of a complete intersection subscheme in $\mathbb{P}^{2n+1}$ of codimension $n+1$ and degree less than $d$ satisfies the variational Hodge conjecture. The interesting part is that we use techniques from commutative algebra to prove this statement (after certain identifications coming from Hodge theory). In the case $n=1$, we use similar methods to characterize all non-reduced components of the Noether-Lefschetz locus of codimension less than or equal to $3d$.
For a fixed $d \ge 5$, the Noether-Lefschetz locus parametrizes smooth degree $d$ surfaces in $\mathbb{P}^3$ with Picard number greater than $1$. This is a countable union of proper algebraic varieties. It is known (due to works of Voisin, Green and others) that the largest irreducible component is of codimension (in the space of all smooth surface in $\mathbb{P}^3$ of degree $d$) equal to $d-3$. The main object of study in this article is: For fixed $r$ greater than $2$ and less than $d$, the locus parametrizing degree $d$ surfaces in $\mathbb{P}^3$ with Picard number at least equal to $r$. It has been conjectured by Griffiths and Harris that the largest component of this locus is of codimension equal to $(r-1)(d-3)-\binom{r-3}{2}$. Furthermore, the irreducible component of this locus parametrizing surfaces with $r$ lines on the same plane is of this codimension. In this article we prove the statement for $d \gg r$.