A specialization inequality for tropical complexes

We prove a specialization inequality relating the dimension of the complete linear series on a variety to the tropical complex of a regular semistable degeneration. Our result extends Baker's specialization inequality to arbitrary dimension.

K3 carpets on minimal rational surfaces and their smoothings

In this paper, we study K3 double structures on minimal rational surfaces [Formula: see text]. The results show there are infinitely many non-split abstract K3 double structures on [Formula: see text] parametrized by [Formula: see text], countably many of which are projective. For [Formula: see text] there exists a unique non-split abstract K3 double structure which is non-projective (see [J.-M. Drézet, Primitive multiple schemes, preprint (2020), arXiv:2004.04921 , to appear in Eur. J. Math.]). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless [Formula: see text] is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on [Formula: see text]. Moreover, we show any embedded projective K3 carpet on [Formula: see text] with [Formula: see text] arises as a flat limit of embeddings degenerating to 2:1 morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on [Formula: see text], embedded by a complete linear series are smooth points if and only if [Formula: see text]. In contrast, Hilbert points corresponding to projective (split) K3 carpets supported on [Formula: see text] and embedded by a complete linear series are always smooth. The results in [P. Bangere, F. J. Gallego and M. González, Deformations of hyperelliptic and generalized hyperelliptic polarized varieties, preprint (2020), arXiv:2005.00342 ] show that there are no higher dimensional analogues of the results in this paper.

Simultaneous percussion by the larvae of a stem-nesting solitary bee – a collaborative defence strategy against parasitoid wasps?

Disturbance sounds to deter antagonists are widespread among insects but have never been recorded for the larvae of bees. Here, we report on the production of disturbance sounds by the postdefecating larva (“prepupa”) of the Palaearctic osmiine bee Hoplitis (Alcidamea) tridentata, which constructs linear series of brood cells in excavated burrows in pithy plant stems. Upon disturbance, the prepupa produces two types of sounds, one of which can be heard up to a distance of 2–3 m (“stroking sounds”), whereas the other is scarcely audible by bare ear (“tapping sounds”). To produce the stroking sounds, the prepupa rapidly pulls a horseshoe-shaped callosity around the anus one to five times in quick succession over the cocoon wall before it starts to produce tapping sounds by knocking a triangularly shaped callosity on the clypeus against the cocoon wall in long uninterrupted series of one to four knocks per second. Sound analysis revealed that the stroking sounds consist of several syllables, which are very similar to the single syllables of the tapping sounds: both last about 0.5 ms and spread over 40 kHz bandwidth from the audible far into the ultrasonic range. The production of stroking sounds by a prepupa induces other prepupae of the same nest to stroke and/or to tap resulting in a long-lasting and simultaneous albeit unsynchronized percussion by numerous prepupae along the whole nest stem. We hypothesize that these disturbance sounds serve an anti-antagonist function and that they have evolved to disturb the reflectance signals that parasitoid wasps use to localize concealed hosts during vibrational sounding.

Divisor Varieties of Symmetric Products

The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and dimension) are smooth, irreducible projective varieties of known dimension. For higher dimensional varieties, the story is less well understood. Our purpose in this paper is to study in detail one class of higher dimensional examples where one can hope for a quite detailed picture, namely (the spaces parametrizing) divisors on the symmetric product of a curve.