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.

Dimension Theory and Degenerations of de Jonquières Divisors

This paper aims at settling the issue of the validity of the de Jonquières formulas. Consider the space of divisors with prescribed multiplicity, or de Jonquières divisors, contained in a linear series on a smooth projective curve. Under the assumption that this space is zero dimensional, the de Jonquières formulas compute the expected number of de Jonquières divisors. Using degenerations to nodal curves we show that, for a general curve equipped with a complete linear series, the space is of expected dimension, which shows that the counts are in fact true. This implies that in the case of negative expected dimension a general linear series on a general curve does not admit de Jonquières divisors of the expected type.

The extremal secant conjecture for curves of arbitrary gonality

We prove the Green–Lazarsfeld secant conjecture [Green and Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), 73–90; Conjecture (3.4)] for extremal line bundles on curves of arbitrary gonality, subject to explicit genericity assumptions.