Bounds on the dimension of linear series on stable curves

We study the dimension of linear series on stable curves. In the first part, we show that a general linear series with semistable multidegree is not special, and obtain results on the dimension of the special loci in the Picard scheme. In the process, we give a new characterization of semistability when the total degree equals the genus of the curve. In the second part, we give a generalization of Clifford's inequality to linear series of uniform multidegree and show that the new bound is achieved on every stable curve.Comment: 23 pages, 4 Figure

Combinatorics of compactified universal Jacobians

We use orientations on stable graphs to express the combinatorial structure of the compactified universal Jacobians in degrees g-1 and g over the moduli space of stable curves, \Mgb, and construct for them graded stratifications compatible with the one of \Mgb. In particular, for a stable curve we exhibit graded stratifications of the compactified Jacobians in terms of totally cyclic, respectively rooted, orientations on subgraphs of its dual graph.Comment: Final version, to appear in Advances in Mathematics. 41 page

On the Severi problem in arbitrary characteristic

We show that Severi varieties parametrizing irreducible reduced planar curves of given degree and geometric genus are either empty or irreducible in any characteristic. As a consequence, we generalize Zariski's theorem to positive characteristic and show that a general reduced planar curve of given geometric genus is nodal. As a byproduct, we obtain the first proof of the irreducibility of the moduli space of smooth projective curves of given genus in positive characteristic, that does not involve a reduction to the characteristic zero case.Comment: 34 pages, 9 figures. Comments are welcome

A Clifford inequality for semistable curves

Let $X$ be a semistable curve and $L$ a line bundle whose multidegree is uniform, i.e., in the range between those of the structure sheaf and the dualizing sheaf of $X$. We establish an upper bound for $h^0(X,L)$, which generalizes the classic Clifford inequality for smooth curves. The bound depends on the total degree of $L$ and connectivity properties of the dual graph of $X$. It is sharp, in the sense that on any semistable curve there exist line bundles with uniform multidegree that achieve the bound.Comment: This was formerly the second part of arxive:2005.12817. It is rewritten to fix a gap in the previous proof. Main results are unchanged, but auxiliary ones have been replaced. 17 pages, 4 Figure

The Inositol- 1,4,5=Trisphosphate System Is Involved in Rapid Effects of Aldosterone in Human Mononuclear Leukocytes

There is increasing evidence for rapid steroid action on electrolyte transport in human mononuclear leukocytes (HML). In HML, aldosterone stimulates the Na+/H+ antiporter within a few minutes. Because a variety of hormones and growth factors activate the Na+/H+ antiporter via protein kinase C and inositol phospholipids, a possible involvement of inositol-1,4,5-trisphosphate (IP3) in the rapid effects of aldosterone in HML was investigated. The stimulation of IP3 generation was started by the addition of aldosterone, concanavalin A, or other steroids. A significant increase in IP3 levels by aldosterone (1 nmol/L, P < 0.05) was found after 1 min, similar to that found after concanavalin A (25 micrograms/mL). Aldosterone caused a concentration-dependent elevation of IP3 levels, with an apparent EC50 of approximately 0.1 nmol/L. Fludrocortisone stimulated IP3 generation at similar concentrations, whereas a weaker IP3 stimulation by glucocorticoids (hydrocortisone, dexamethasone) occurred at micromolar concentrations only. Canrenone, a potent inhibitor of classical aldosterone action, was not effective up to a concentration of 100 nmol/L. These findings show kinetic and pharmacological similarities with both the functional data on Na+/H+ antiport stimulation by aldosterone and the studies of 125I-aldosterone binding to plasma membranes of HML. Thus, these data are the first to indicate an involvement of the phosphoinositide pathway in the rapid membrane effects of aldosterone

Degeneration of curves on some polarized toric surfaces

We address the following question: Given a polarized toric surface (S,L), and a general integral curve C of geometric genus g in the linear system |L|, do there exist degenerations of C in |L| to general integral curves of smaller geometric genera? We give an affirmative answer to this question for surfaces associated to h-transverse polygons, provided that the characteristic of the ground field is large enough. We give examples of surfaces in small characteristic, for which the answer to the question is negative. In case the answer is affirmative, we deduce that a general curve C as above is nodal. In characteristic 0, we use the result to show irreducibility of Severi varieties of a large class of polarized toric surfaces with h-transverse polygon.Comment: v1: 34 pages, 11 figures. v2: improved the bound on the characteristic, the result is now sharp for characteristic at least 3. 35 pages, 12 figures. Comments are welcome

Correction: On the Severi problem in arbitrary characteristic

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Compactified Jacobians as Mumford models

We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g, as constructed by Caporaso, Esteves, and Simpson, are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that the Simpson compactified Jacobian in degree g is induced by the tropical break divisor decomposition.Comment: 23 page