Second fundamental form of the Prym map in the ramified case

In this paper we study the second fundamental form of the Prym map $P_{g,r}: R_{g,r} \rightarrow {\mathcal A}^{\delta}_{g-1+r}$ in the ramified case $r>0$. We give an expression of it in terms of the second fundamental form of the Torelli map of the covering curves. We use this expression to give an upper bound for the dimension of a germ of a totally geodesic submanifold, and hence of a Shimura subvariety of ${\mathcal A}^{\delta}_{g-1+r}$, contained in the Prym locus.Comment: To appear in Galois Covers, Grothendieck-Teichmueller Theory and Dessins d'Enfants - Interactions between Geometry, Topology, Number Theory and Algebra. Springer Proceedings in Mathematics & Statistics. arXiv admin note: text overlap with arXiv:1711.0342

On totally geodesic submanifolds in the Jacobian locus

We study submanifolds of A_g that are totally geodesic for the locally symmetric metric and which are contained in the closure of the Jacobian locus but not in its boundary. In the first section we recall a formula for the second fundamental form of the period map due to Pirola, Tortora and the first author. We show that this result can be stated quite neatly using a line bundle over the product of the curve with itself. We give an upper bound for the dimension of a germ of a totally geodesic submanifold passing through [C] in M_g in terms of the gonality of C. This yields an upper bound for the dimension of a germ of a totally geodesic submanifold contained in the Jacobian locus, which only depends on the genus. We also study the submanifolds of A_g obtained from cyclic covers of the projective line. These have been studied by various authors. Moonen determined which of them are Shimura varieties using deep results in positive characteristic. Using our methods we show that many of the submanifolds which are not Shimura varieties are not even totally geodesic.Comment: To appear on International Journal of Mathematic

Single-species fragmentation: the role of density-dependent feedbacks

Internal feedbacks are commonly present in biological populations and can play a crucial role in the emergence of collective behavior. We consider a generalization of Fisher-KPP equation to describe the temporal evolution of the distribution of a single-species population. This equation includes the elementary processes of random motion, reproduction and, importantly, nonlocal interspecific competition, which introduces a spatial scale of interaction. Furthermore, we take into account feedback mechanisms in diffusion and growth processes, mimicked through density-dependencies controlled by exponents $\nu$ and $\mu$, respectively. These feedbacks include, for instance, anomalous diffusion, reaction to overcrowding or to rarefaction of the population, as well as Allee-like effects. We report that, depending on the dynamics in place, the population can self-organize splitting into disconnected sub-populations, in the absence of environment constraints. Through extensive numerical simulations, we investigate the temporal evolution and stationary features of the population distribution in the one-dimensional case. We discuss the crucial role that density-dependency has on pattern formation, particularly on fragmentation, which can bring important consequences to processes such as epidemic spread and speciation

Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the case $g'=0$. Here we find 6 families of Galois coverings, all with $g' = 1$ and $g=2,3,4$ and we show that these are the only families with $g'=1$ satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of $\mathsf{A}_g$, while the other examples arise from certain Shimura subvarieties of $\mathsf{A}_g$ already obtained as families of Galois coverings of $\mathbb{P}^1$ in [FGP]. Finally we prove that if a family satisfies this sufficient condition with $g'\geq 1$, then $g \leq 6g'+1$.Comment: 18 pages, to appear in Geometriae Dedicat

Satellite To Satellite Doppler Tracking (SSDT) for mapping of the Earth's gravity field

Two SSDT schemes were evaluated: a standard, low-low, SSDT configuration, which both satellites are in basically the same low altitude nearly circular orbit and the pair is characterized by small angular separation; and a more general configuration in which the two satellites are in arbitrary orbits, so that different configurations can be comparatively analyed. The standard low-low SSDT configuration is capable of recovering 1 deg X 1 deg surface anomalies with a strength as low as 1 milligal, located on the projected satellite path, when observing from a height as large as 300 km. The Colombo scheme provides an important complement of SSDT observations, inasmuch as it is sensitive to radial velocity components, while keeping at the same performance level both measuring sensitivity and measurement resolution

On the first Gaussian map for Prym-canonical line bundles

We prove by degeneration to Prym-canonical binary curves that the first Gaussian map of the Prym canonical line bundle $\omega_C \otimes A$ is surjective for the general point [C,A] of R_g if g >11, while it is injective if g < 12.Comment: To appear in Geometriae Dedicata. arXiv admin note: text overlap with arXiv:1105.447