arXiv subjects
Publications and source records attributed to Abel Castorena.
Let $X$ be a projective bundle over a smooth curve $C$ of genus $g \ge 3$, and consider the relative hyperplane bundle ${\mathcal O}_X (1) \to X$. Let $V \subseteq H^0 ( X , {\mathcal O}_X (1) )$ be a generating subspace. We prove that when $C$, $X$ and $V$ are general in moduli and ${\mathcal O}_X (1)$ is sufficiently ample, the kernel bundle of the system $({\mathcal O}_X (1) , V)$ is ${\mathcal O}_X (1)$-stable.
Let $X\subseteq \mathbb{P}^{3}$ be a very general element of the Noether-Lefschetz divisor that parametrizing smooth quartic surfaces containing a line. Let $L\subseteq X$ denote the corresponding line. We study the curves contained in $X$ and analyze their behavior in the Hilbert scheme. We first determine which linear systems contain smooth irreducible curves. For most classes, we verify that the general member is a smooth point of the expected Hilbert scheme. Finally we compute the Rao function of any curve on $X$.
We classify ACM curves contained in a surface of degree d in $\mathbb{P}^{3}$ in terms of weak admissible pairs. In the case of a very general smooth determinantal quartic surface, we provide a geometric description of these curves and compute their Picard classes on the surface. Finally, we present a generalization to ACM closed subvarieties of codimension $1$ on a hypersurface in $\mathbb{P}^{n}$.
Let $E$ be a vector bundle over a smooth curve $C$, and $V$ a generating space of sections of $E$. We characterise Mumford linear stability of the associated projective model of $\mathbb{P} E^\vee$ in $\mathbb{P} V^\vee$ in terms of geometric and cohomological properties of the coherent system $(E, V)$, and give some applications. We show that any $\mathbb{P}^{r-1}$-bundle over $C$ has a linearly stable model in $\mathbb{P}^{n-1}$ for any $n \ge r+2$. Furthermore; linear stability of $(E, V)$ is a necessary condition for stability of the kernel bundle $M_{E, V}$ of $(E, V)$, which is predicted by Butler's conjecture for general $C$ and $(E, V)$. We give new examples showing that it is not in general sufficient; in particular, a general bundle $E$ of large degree fits into a linearly stable coherent system $(E, V)$ with nonsemistable kernel bundle. Finally, we use these ideas to show the stability of $M_{E, V}$ for certain $(E, V)$ of type $(r, d, r+2)$ where $E$ is not necessarily stable.
Let $f\in\mathbb{C}[x_1,\ldots,x_s]$ be a weighted homogeneous polynomial having an isolated singularity and $\mathcal{T}_n(f)$ be its higher Nash blowup local algebra. We show that $\mathcal{T}_n(f)$ does not admit negative weighted derivations for $n\geq2$. This answers affirmatively a conjecture of Hussain-Ma-Yau-Zuo.
There is a well studied notion of GIT-stability for coherent systems over curves, which depends on a real parameter $\alpha$. For generated coherent systems, there is a further notion of stability derived from Mumford's definition of linear stability for varieties in projective space. Let $\alpha_S$ be close to zero and $\alpha_L \gg 0$. We show that a generated coherent system which is $\alpha_S$-stable and linearly stable is $\alpha_L$-stable, and give examples showing that without further assumptions, there are no other implications between these three types of stability. We observe that several of the systems constructed have stable dual span bundle, including systems which are not $\alpha$-semistable for any value of $\alpha$. We use this to prove a case of Butler's conjecture for systems of type $(2, d, 5)$.
We study moduli stacks of principal $\Bbb C^*$-bundles over nodal complex algebraic curves and determine their rational cohomology algebras in terms of Chern classes.
We study the various arithmetic and geometric Frobenius morphisms on the moduli stack of principal bundles over a smooth projective algebraic curve and determine explicitly their actions on the $\ell-$adic cohomology of the moduli stack in terms of Chern classes.
The notion of linear stability of a variety in projective space was introduced by Mumford in the context of GIT. It has subsequently been applied by Mistretta and others to Butler's conjecture on stability of the dual span bundle (DSB) $M_{V, E}$ of a general generated coherent system $( E, V )$. We survey recent progress in this direction on rank one coherent systems, prove a new result for hyperelliptic curves, and state some open questions. We then extend the definition of linear stability to generated coherent systems of higher rank. We show that various coherent systems with unstable DSB studied by Brambila-Paz, Mata-Gutierrez, Newstead and Ortega are also linearly unstable. We show that linearly stable coherent systems of type $(2, d, 4)$ for low enough $d$ have stable DSB, and use this to prove a particular case of a generalized Butler conjecture. We then exhibit a linearly stable generated coherent system with unstable DSB, confirming that linear stability of $( E, V )$ in general remains weaker than semistability of $M_{V, E}$ in higher rank. We end with a list of open questions on the higher rank case.
We prove that the ideal generated by the maximal minors of the higher-order Jacobian matrix of a weighted homogeneous polynomial is also weighted homogeneous. As an application, we give a partial answer to a conjecture concerning the non-existence of negative weight derivations on the higher Nash blowup local algebra of a hypersurface.
Applying Geometric Invariant Theory (GIT), we study the stability of foliations of degree 3 on P^2 with a unique singular point of multiplicity 1, 2, or 3 and Milnor number 13. In particular, we characterize those foliations for multiplicity 2 in three cases: stable, strictly semistable, and unstable.
In this paper we give the first steps toward the study of the Harbourne-Hirschowitz condition and the Anticanonical Orthogonal Property for regular surfaces. To do so, we consider the Kodaira dimension of the surfaces and study the cases based on the Enriques-Kodaira classification.
In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in $\mathbb{P}^5$. We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge's curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-1}{2}$ and the moduli space of Humbert-Edge's curves of type $n\geq 5$ where $n$ is an odd number.
The main goal of this article is to compute the class of the divisor of $\overline{\mathcal{M}}_3$ obtained by taking the closure of the image of $Ω\mathcal{M}_3(6;-2)$ by the forgetful map. This is done using Porteous formula and the theory of test curves. For this purpose, we study the locus of meromorphic differentials of the second kind, computing the dimension of the map of these loci to $\mathcal{M}_g$ and solving some enumerative problems involving such differentials in low genus. A key tool of the proof is the compactification of strata recently introduced by Bainbridge-Chen-Gendron-Grushevsky-Möller.
In previous works, the authors investigated the relationships between linear stability of a generated linear series $|V|$ on a curve $C$, and slope stabillity of the vector bundle $M_{V,L} := \ker (V \otimes \mathcal{O}_C \to L)$. In particular, the second named author and L. Stoppino conjecture that, for a complete linear system $|L|$, linear (semi)stability is equivalent to slope (semi)stability of $M_V$, and the first and third named authors proved that this conjecture holds for hyperelliptic and for generic curves. In this work we provide a counterexample to this conjecture on any smooth plane curve of degree $7$.
Let $C$ be a smooth complex irreducible projective curve of genus $g$ with general moduli, and let $(L,H^0(L))$ be a generated complete linear series of type $(d,r+1)$ over $C$. The syzygy bundle, denoted by $M_L$, is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_C\to L$. In this work we have a double purpose. The first one is to give new examples of stable syzygy bundles admitting theta divisor over general curves. We prove that if $M_L$ is strictly semistable then $M_L$ admits reducible theta divisor. The second purpose is to study the cohomological semistability of $M_L$, and in this direction we show that when $L$ induces a birational map, the syzygy bundle $M_L$ is cohomologically semistable, and we obtain precise conditions for the cohomological semistability of $M_L$ where such conditions agree with the semistability conditions for $M_L$.
In this work we describe a construction of semistable fibrations over an elliptic curve with one unique singular fibre and we give effective examples using monodromy of curves.
For a general curve C of genus $g\geq 10$, we show that the Brill- Noether locus $B^4(2,K_C)$ contains irreducible sub-varieties $B_3\supset B_4 \supset \cdots \supset B_n$, where $B_n$ is of dimension $3g-10-n$ and $B_3$ is an irreducible component of the expected dimension $3g-13$.