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Juliana Coelho

Publications and source records attributed to Juliana Coelho.

11 recordsLinked to original sources
arXiv2026

Type of homomorphisms of complex tori

Using determinantal divisors of integral matrices, that is, greatest common divisors of minors of fixed order, we introduce the notion of type of a homomorphism $f$ of complex tori, which is similar to the type of a polarization. We show that the type is invariant under composition with isomorphisms, and that it completely describes the kernel of $f$ as a group. More precisely, the quotient of the kernel by its connected component containing 0 is a product of cyclic groups whose orders are determined by the type of $f$. Since the type can be computed from any rational representation of $f$, this gives an effective way to determine the kernel of a homomorphism. As a consequence, we compute the classic invariants degree and exponent of $f$, when $f$ has finite kernel. When $f$ is an isogeny, we also compute the type of its inverse isogeny from that of $f$. Finally, we compare the type of a polarization to the type of its associated isogeny.

math.RA↗
arXiv2023

Gonality of curves whose normalizations are one or two copies of $\mathbb P^1$

We study the gonality of curves $C$ over $\mathbb C$ whose normalization is composed of one or two copies of $\mathbb P^1$. In the first case, $C$ is a nodal curve with $g(C)$ nodes, and in the second case $C$ is a so-called binary curve. In any case we show that the usual bound $\mathrm{gon}(C)\leq\lfloor\frac{g(C)+3}{2}\rfloor$ holds if $g(C)\geq 2$, with equality holding generically.

math.AG↗
arXiv2020

Characterizing gonality for two-component stable curves

It is a well-known result that a stable curve of compact type over $\mathbb{C}$ having two components is hyperelliptic if and only if both components are hyperelliptic and the point of intersection is a Weierstrass point for each of them. With the use of admissible covers, we generalize this characterization in two ways: for stable curves of higher gonality having two smooth components and one node; and for hyperelliptic and trigonal stable curves having two smooth non rational components and any number of nodes.

math.AG↗
arXiv2020

Abel-Prym maps for isotypical components of Jacobians

Let $C$ be a smooth non rational projective curve over the complex field $\mathbb{C}$. If $A$ is an abelian subvariety of the Jacobian $J(C)$, we consider the Abel-Prym map $φ_A : C \rightarrow A$ defined as the composition of the Abel map of $C$ with the norm map of $A$. The goal of this work is to investigate the degree of the map $φ_A$ in the case where $A$ is one of the components of an isotypical decomposition of $J(C)$. In this case we obtained a lower bound for $\mathrm{deg}(φ_A)$ and, under some hypotheses, also an upper bound. We then apply the results obtained to compute degrees of Abel-Prym maps in four cases. In particular, these examples show that both bounds are sharp.

math.AG↗
arXiv2019

On the gonality of stable curves

In this paper we use admissible covers to investigate the gonality of a stable curve $C$ over $\mathbb{C}$. If $C$ is irreducible, we compare its gonality to that of its normalization. If $C$ is reducible, we compare its gonality to that of its irreducible components. In both cases we obtain lower and upper bounds. Furthermore, we show that four admissible covers constructed give rise to generically injective maps between Hurwitz schemes. We show that the closures of the images of three of these maps are components of the boundary of the target Hurwitz schemes, and the closure of the image of the remaining map is a component of a certain codimension-1 subscheme of the boundary of the target Hurwitz scheme.

math.AG↗
arXiv2013

On the geometry of Abel maps for nodal curves

In this paper we give local conditions to the existence of Abel maps for nodal curves that are limits of Abel maps for smooth curves. We use this result to construct Abel maps for any degree for nodal curves with two components.

math.AG↗
arXiv2012

Degree-2 Abel maps for nodal curves

We present numerical conditions for the existence of natural degree-2 Abel maps for any given nodal curve. Cocoa scripst were written and have so far verified the validity of the conditions for numerous curves.

math.AG↗
arXiv2011

Brill-Noether locus of rank 1 and degree g-1 on a nodal curve

In this paper we consider the Brill-Noether locus $W_{\underline d}(C)$ of line bundles of multidegree $\underline d$ of total degree $g-1$ having a nonzero section on a nodal reducible curve $C$ of genus $g\geq2$. We give an explicit description of the irreducible components of $W_{\underline d}(C)$ for a semistable multidegre $\underline d$. As a consequence we show that, if two semistable multidegrees of total degre $g-1$ on a curve with no rational components differ by a twister, then the respective Brill-Noether loci have isomorphic components.

math.AG↗
arXiv2009

Abel maps for curves of compact type

Recently, the first Abel map for a stable curve of genus g>1 has been constructed. Fix an integer d>0 and let C be a stable curve of compact type of genus g>1. We construct two d-th Abel maps for C, having different targets, and we compare the fibers of the two maps. As an application, we get a characterization of hyperelliptic stable curves of compact type with two components via the 2-nd Abel map.

math.AG↗
arXiv2007

Abel maps of Gorenstein curves

For a Gorenstein curve X and a nonsingular point P of X, we construct Abel maps A from X to J_X^1 and A_P from X to J_X^0, where J_X^i is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A_P are shown to have the same arithmetic genus of X. Also, A and A_P are shown to be embeddings away from rational subcurves L of X meeting the closure of X-L in separating nodes. Finally, we establish a connection with Seshadri's moduli scheme U_X(1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U_X(1).

math.AG↗