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We study the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})from an\mathbf{A}^{1}-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme\mathrm{Hilb}_{d}(\mathbf{A}^{\infty})is\mathbf{A}^{1}-equivalent to the Grassmannian of(d-1)-planes in\mathbf{A}^{\infty}. We then describe the\mathbf{A}^{1}-homotopy type of\mathrm{Hilb}_{d}(\mathbf{A}^{n})in a certain range, fornlarge compared tod. For example, we compute the integral cohomology of\mathrm{Hilb}_{d}(\mathbf{A}^{n})(\mathbf{C})in a range. We also deduce that the forgetful map\mathcal{FF}\mathrm{lat}\to\mathcal{V}\mathrm{ect}from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an\mathbf{A}^{1}-equivalence after group completion. This implies that the moduli stack\mathcal{FF}\mathrm{lat}, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum\mathrm{kgl}representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the\mathrm{kgl}-homology of smooth proper schemes over a perfect field. 

We show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some classification results on varieties admitting endomorphisms (for Fano threefolds of Picard number one and several other cases) to any characteristic. The classification results in characteristic zero are due to Amerik–Rovinsky–Van de Ven, Hwang–Mok, Paranjape–Srinivas, Beauville, and Shao–Zhong. Our method also bounds the degree of morphisms into a given variety. Finally, we relate endomorphisms to global $F$-regularity. 

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that the higher cohomology of every bundle of differential forms tensored with an ample line bundle is zero. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano threefolds that satisfy Bott vanishing. There are many more than expected. 

Abstract Our ability to forecast the spatial and temporal patterns of ecological processes at continental scales has drastically improved over the past decade. Yet, predicting ecological patterns at broad scales while capturing fine-scale processes is a central challenge of ecological forecasting given the inherent tension between grain and extent, whereby enhancing one often diminishes the other. We leveraged 10 years of terrestrial and atmospheric data (2012–2021) to develop a high-resolution (2.9 × 2.9 km), radar-driven bird migration forecast model for a highly active region of the Mississippi flyway. Based on the suite of candidate models we examined, adding terrestrial predictors improved model performance only marginally, whereas spatially distant atmospheric predictors, particularly air temperature and wind speed from focal and distant regions, were major contributors to our top model, explaining 56% of variation in regional migration activity. Among terrestrial predictors, which ranked considerably lower than atmospheric predictors in terms of variable importance, vegetation phenology, artificial light at night, and percent of forest cover were the most important predictors. Furthermore, we scale this model to demonstrate the capacity to generate real-time, high-resolution forecasts for the continental United States that explained up to 65% of national variation. Our study demonstrates an approach for increasing the resolution of migration forecasts, which could facilitate the integration of radar with other data sources and inform dynamic conservation efforts at a local scale that is more relevant to threats, such as anthropogenic light at night. 

Abstract The major sources of macronutrients (nitrate, ammonium, phosphate, and silicic acid) in Jakolof Bay, Alaska are submarine groundwater discharge (SGD), rivers, and offshore water. We estimated SGD using natural geochemical tracers (radon and radium), a salt mass balance, and a two-component salinity mixing equation based on the change in groundwater salinity on falling lower low tide. Previous studies have hypothesized that the major macronutrient input into Jakolof Bay is offshore water. This study challenges that assumption by determining the relative contribution of macronutrients from SGD relative to offshore water and rivers. Here, SGD is tidally driven and, as the Northern Gulf of Alaska experiences some of the largest tidal ranges in the world, the SGD fluxes from this region are high relative to the global average regardless of local sediment type. The fluxes ranged from 596 ± 85 cm day⁻¹at low tide to 97 ± 83 cm day⁻¹at high tide and are predominantly composed of recirculated seawater (89%) rather than freshwater (11%). The major macronutrients in seawater had different input mechanisms into the semi-enclosed bay. SGD and offshore waters contend as the primary sources of nitrate, which is shown to be the limiting nutrient in this coastal area, while SGD dominates the input of silicic acid. Conversely, the aquifer is found to be a sink for phosphate, indicating that the nutrient is primarily sourced from offshore water. 

The Line Height Absorption (LHA) method uses absorption of light to estimate chlorophyll-a. While most users consider regional variability and apply corrections, the effect of temporal variability is typically not explored. The Northern Gulf of Alaska (NGA) was selected for this study because there was no published regional value and its large swings in temporal productivity would make it a good candidate to evaluate the effect of temporal variability on the relationship. The mean NGA value of 0.0114 obtained here should be treated with caution, as variation in the slope of the relationship (a_LH*), and thus chlorophyll-a estimates, in the NGA region varied by ∼25% between spring (a_LH* = 0.0109) and summer (a_LH* = 0.0137). Results suggest that this change is driven by a shift in pigment packaging and cell size associated with changes in mixed layer depth and stratification. Consideration of how temporal variability may affect the accuracy of the LHA method in other regions is thus recommended. 

We construct klt projective varieties with ample canonical class and the smallest known volume. We also find exceptional klt Fano varieties with the smallest known anti-canonical volume. We conjecture that our examples have the smallest volume in every dimension, and we give low-dimensional evidence for that. In order to improve on earlier examples, we are forced to consider weighted hypersurfaces that are not quasi-smooth. We show that our Fano varieties are exceptional by computing their global log canonical threshold (or alpha-invariant) exactly; it is extremely large, roughly 2^{2^n} in dimension n. These examples give improved lower bounds in Birkar’s theorem on boundedness of complements for Fano varieties.