Quot schemes and Fourier-Mukai transformation
We consider several related examples of Fourier-Mukai transformations involving the quot scheme. A method of showing conservativity of these Fourier-Mukai transformations is described.
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arXiv subjects
A. J. Parameswaran
Publications and source records attributed to A. J. Parameswaran.
Milnor and Tjurina numbers for an isolated complete intersection singularity
This paper aims to prove that given a isolated complete intersection singularity, the Milnor number will be bounded by a bound depending only on Tjurina number and dimension of the singularity. The proof uses A$\mathfrak{m}$AC (introduced in arXiv:2204.05594) and as with such methods, the bound is purely existential.
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Genuinely ramified maps and pseudo-stable vector bundles
Let $X$ and $Y$ be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let $f : Y \rightarrow X$ be a finite generically smooth morphism such that the corresponding homomorphism between the étale fundamental groups $f_*:π^{\rm et}_{1}(Y) \rightarrowπ^{\rm et}_{1}(X)$ is surjective. Fix a polarization on $X$ and equip $Y$ with the pulled back polarization. For a point $y_0\in Y$, let $\varpi(Y, y_0)$ (respectively, $\varpi(X, f(y_0))$) be the affine group scheme given by the neutral Tannakian category defined by the strongly pseudo-stable vector bundles of degree zero on $Y$ (respectively, $X$). We prove that the homomorphism $\varpi(Y, y_0) \rightarrow \varpi(X, f(y_0))$ induced by $f$ is surjective. Let $E$ be a pseudo-stable vector bundle on $X$ and $F \subset f^*E$ a pseudo-stable subbundle with $μ(F)= μ(f^*E)$. We prove that $f^*E$ is pseudo-stable and there is a pseudo-stable subbundle $W \subset E$ such that $f^*W = F$ as subbundles of $f^*E$.
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Stability and semi-stability of (2,2)-type surfaces
We describe the GIT compactification of the moduli of (2,2)-type effective divisors of $\mathbb{P}^1\times\mathbb{P}^2$ (i.e., surfaces of the linear system $\vert π_1^*\mathcal{O}_{\mathbb{P}^1}(2)\otimes π_2^*\mathcal{O}_{\mathbb{P}^2}(2)\vert$ ) which are generically Del Pezzo surfaces of degree two. In order to get the compactification, we characterize stable and semi-stable (2,2)-type surfaces, and also determine the equivalence classes of strictly semi-stable (2,2)-type surfaces. Moreover, we describe the boundary of the moduli of (2,2)-type surfaces.
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Connections and genuinely ramified maps of curves
Given a singular connection $D$ on a vector bundle $E$ over an irreducible smooth projective curve $X$, defined over an algebraically closed field, we show that there is a unique maximal subsheaf of $E$ on which $D$ induces a nonsingular connection. Given a generically smooth map $ϕ: Y \rightarrow\ X$ between irreducible smooth projective curves, and a singular connection $(V, D)$ on $Y$, the direct image $ϕ_*V$ has a singular connection. Let $\textbf{R}(ϕ_*{\mathcal O}_Y)$ be the unique maximal subsheaf on which the singular connection on $ϕ_*{\mathcal O}_Y$ -- corresponding to the trivial connection on ${\mathcal O}_Y$ -- induces a nonsingular connection. We prove that the homomorphism of étale fundamental groups $ϕ_*: π_1^{\rm et}(Y, y_0) \rightarrow π_1^{\rm et}(X, ϕ(y_0))$ induced by $ϕ$ is surjective if and only if ${\mathcal O}_X \subset \textbf{R}(ϕ_*{\mathcal O}_Y)$ is the unique maximal semistable subsheaf. When the characteristic of the base field is zero, this homomorphism $ϕ_*$ is surjective if and only if ${\mathcal O}_X = \textbf{R}(ϕ_*{\mathcal O}_Y)$. For any nonsingular connection $D$ on a vector bundle $V$ over $X$, there is a natural map $V\hookrightarrow {\bf R}(ϕ_*ϕ^*V)$. When the characteristic of the base field is zero, we prove that the map $ϕ$ is genuinely ramified if and only if $V ={\bf R}(ϕ_*ϕ^*V)$.
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On the stability of pulled back parabolic vector bundles
Take an irreducible smooth projective curve $X$ defined over an algebraically closed field of characteristic zero, and fix finitely many distinct point $D\, =\, \{x_1,\, \cdots,\, x_n\}$ of it; for each point $x\, \in\, D$ fix a positive integer $N_x$. Take a nonconstant map $f\, :\, Y\, \longrightarrow \, X$ from an irreducible smooth projective curve. We construct a natural subbundle $\mathcal{F}\, \subset\, f_*{\mathcal O}_Y$ using $(D,\, \{N_x\}_{x\in D})$. Let $E_*$ be a stable parabolic vector bundle whose parabolic weights at each $x\, \in\, D$ are integral multiples of $\frac{1}{N_x}$. We prove that the pullback $f^*E_*$ is also parabolic stable, if ${\rm rank}(\mathcal{F})\,=\, 1$.
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Jacobians, Anti-affine groups and torsion points
We give criteria for the Jacobian of a singular curve $X$ with at most ordinary $n$-point singularities to be anti-affine. In particular, for the case of curves with single ordinary double point we exhibit a relation with torsion divisors. If the geometric genus of the singular curve is atleast 3 and the normalization is non-hyperelliptic and non-bielliptic, then except for finitely many cases the Jacobian of $X$ is anti-affine. Furthermore, if the normalization is a general curve of genus atleast 3 then the Jacobian of $X$ is always anti-affine.
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Genuinely ramified maps and stability of pulled-back parabolic bundles
Let $f : X \rightarrow Y$ be a genuinely ramified map between irreducible smooth projective curves defined over an algebraically closed field. Let $P$ be a branch data on $Y$ such that $P(y)$ and $B_f(y)$ where $B_f$ is branch data for $f$ are linearly disjoint for every $y \in, Y$. Further assume that either $P$ is tame or $f$ is Galois. Then the pullback, by $f$, of any stable parabolic bundle on $Y$ with respect to $P$ is actually a stable parabolic bundle on $X$ with respect to $f^*P$.
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Genuinely ramified maps and stable vector bundles
Let $f : X \rightarrow Y$ be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between étale fundamental groups $f_* : π_1^{\rm et}(X)\rightarrowπ_1^{\rm et}(Y)$ is surjective. Fix a polarization on $Y$ and equip $X$ with the pullback, by $f$, of this polarization on $Y$. Given a stable vector bundle $E$ on $X$, we prove that there is a vector bundle $W$ on $Y$ with $f^*W$ isomorphic to $E$ if and only if the direct image $f_*E$ contains a stable vector bundle $F$ such that $$ \frac{{\rm degree}(F)}{{\rm rank}(F)}= \frac{1}{{\rm degree}(f)}\cdot \frac{{\rm degree}(E)}{{\rm rank}(E)} $$ We also prove that $f^*V$ is stable for every stable vector bundle $V$ on $Y$.
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Ulrich line bundles on double planes
Consider a smooth complex surface $X$ which is a double cover of the projective plane $\mathbb{P}^2$ branched along a smooth curve of degree $2s$. In this article, we study the geometric conditions which are equivalent to the existence of Ulrich line bundles on $X$ with respect to this double covering. Also, for every $s\geq 1$, we describe the classes of such surfaces which admit Ulrich line bundles and give examples.
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Asymptotic slopes and strong semistability on surfaces
In this article we study asymptotic slopes of strongly semistable vector bundles on a smooth projective surface. A connection between asymptotic slopes and strong restriction theorem of a strongly semistable vector bundle is shown. We also give an equivalent criterion of strong semistability of a vector bundle in terms of its asymptotic slopes under some assumptions on the surface and on the bundle.
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Higher dimensional formal orbifolds and orbifold bundles in positive characteristic
In \cite{KP}, the last two authors introduced formal orbifold curves defined over an algebraically closed field of positive characteristics. They studied both étale and Nori fundamental group schemes associated to such objects. Our aim here is to study the higher dimensional analog of these objects objects and their fundamental groups.
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Lefschetz Theorem Using Real Morse Theory
We prove the Lefschetz hyperplane section theorem using a simpler machinery by making the observation that we can compose the Lefschetz Pencil with a Real Morse function to get a map from the variety to $\mathbb{R}$ which is "close" to being a Real Morse function. The proof uses a new method unlike the conventional one which uses vanishing cycles, thimbles and monodromy. We prove the genus formula for plane curves using Morse theory, Lefschetz pencil and Bézout's theorem. And then we prove the Riemann Hurwitz formula for ramified maps between curves by employing techniques from deformation theory. Lastly, we prove the Lefschetz Hyperplane Section Theorem solely using Real Morse Theory and exact sequences.
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Canonical subsheaves of torsionfree semistable sheaves
Let $F$ be a torsionfree semistable coherent sheaf on a polarized normal projective variety. We prove that $F$ has a unique maximal locally free subsheaf $V$ such that $F/V$ is torsionfree and $V$ admits a filtration of subbundles for which each successive quotient is stable whose slope is $μ(F)$. We also prove that $F$ has a unique maximal reflexive subsheaf $W$ such that $F/W$ is torsionfree and $W$ admits a filtration of subsheaves for which each successive quotient is a stable reflexive sheaf whose slope is $μ(F)$. We show that these canonical subsheaves behave well with respect to the pullback operation by étale Galois covering maps. Given a separable finite surjective map $ϕ:Y \, \rightarrow X$ between normal projective varieties, we give a criterion for the induced homomorphism of étale fundamental groups $ϕ_* : π^{\rm et}_{1}(Y) \rightarrow π^{\rm et}_{1}(X)$ to be surjective; this criterion is in terms of the above mentioned unique maximal locally free subsheaf associated to $ϕ_*{\mathcal O}_Y$.
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Ramified covering maps and stability of pulled back bundles
Let $f:C\rightarrow D$ be a nonconstant separable morphism between irreducible smooth projective curves defined over an algebraically closed field. We say that $f$ is genuinely ramified if ${\mathcal O}_D$ is the maximal semistable subbundle of $f_*{\mathcal O}_C$ (equivalently, the homomorphism of etale fundamental groups is surjective). We prove that the pullback $f^*E\rightarrow C$ is stable for every stable vector bundle $E$ on $D$ if and only if $f$ is genuinely ramified.
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Etale Fundamental group of moduli of torsors under Bruhat-Tits group scheme over a curve
Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a Bruhat-Tits group scheme on $X$ which is generically semi-simple and trivial. We show that the étale fundamental group of the moduli stack $\mathcal{M}_X(\mathcal{G})$ of torsors under $\mathcal{G}$ is isomorphic to that of the moduli stack $\mathcal{M}_X(G)$ of principal $G$-bundles. For any smooth, noetherian and irreducible stack $\mathcal{X}$, we show that an inclusion of an open substack $\mathcal{X}^\circ$, whose complement has codimension at least two, will induce an isomorphism of étale fundamental group. Over $\mathbb{C}$, we show that the open substack of regularly stable torsors in $\mathcal{M}_X(\mathcal{G})$ has complement of codimension at least two when $g_X \geq 3$. As an application, we show that the moduli space $M_X(\mathcal{G})$ of $\mathcal{G}$-torsors is simply-connected.
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Brauer group of punctual Quot scheme of points on a smooth projective surface
Let $X$ be a smooth projective surface over an algebraically closed field $k$ such that $char(k) \neq 2$. Let $X^{[d]}$ denote the punctual Hilbert scheme of zero dimensional quotients of degree $d$ and $X^{(d)}$ denote the symmetric product of $X$. For $\ell \neq 2$, we give a formula for the $\ell$-primary part of the Brauer group of $X^{[2]}$. We show that the Hilbert to Chow morphism induces an isomorphism of cohomological Brauer groups for $d=2$ and a similar result for $d \geq 3$. Let $Q(r,d)$ denote the punctual Quot-scheme parametrising zero dimensional quotients of $\mathcal{O}_X^{ \oplus r}$ of degree $d$. We show that the natural morphism from $Q(r,d) \rightarrow X^{[d]}$ induces an isomorphism on cohomological Brauer groups.
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Thom irregularity and Milnor tube fibrations
We find natural and convenient conditions which allow us to produce classes of genuine real map germs with Milnor tube fibration, either with Thom regularity or without it.
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