Tropical Severi Varieties

We study the tropicalizations of Severi varieties, which we call tropical Severi varieties. In this paper, we give a partial answer to the following question, ``describe the tropical Severi varieties explicitly.'' We obtain a description of tropical Severi varieties in terms of regular subdivisions of polygons. As an intermediate step, we construct explicit parameter spaces of curves. These parameter spaces are much simpler objects than the corresponding Severi variety and they are closely related to flat degenerations of the Severi variety, which in turn describes the tropical Severi variety. As an application, we understand G.Mikhalkin's correspondence theorem for the degrees of Severi varieties in terms of tropical intersection theory. In particular, this provides a proof of the independence of point-configurations in the enumeration of tropical nodal curves.Comment: 25 pages, Final version accepted to Portugal. Mat

The Stock of Private Real Estate Capital in U.S. Metropolitan Areas

In this paper, we describe the construction of estimates of private real estate capital for each of 242 MSAs, annually, for 1982 through 1994. We compute three such series: (1) total private real estate capital (residential and nonresidential); (2) private single-family residential capital; and (3) private income property capital (multifamily housing plus nonresidential real estate , or (1) less (2)). We then model the determinants of each series, and use the results to predict the value of the capital stock for a larger set of 295 MSAs.

Constraints on Inflaton-Higgs Couplings

According to the best-fit parameters of the Standard Model, the Higgs field's potential reaches a maximum at a field value $h \sim 10^{10-11}$ GeV and then turns over to negative values. During reheating after inflation, resonance between the inflaton and the Higgs can cause the Higgs to fluctuate past this maximum and run down the dangerous side of the potential if these fields couple too strongly. In this paper, we place constraints on the inflaton-Higgs couplings such that the probability of the Higgs entering the unstable regime during reheating is small. To do so, the equations of motion are approximately solved semi-analytically, then solved fully numerically. Next the growth in variance is used to determine the parameter space for $\kappa$ and $\alpha$, the coupling coefficients for inflaton-Higgs cubic and quartic interactions, respectively. We find the upper bounds of $\kappa < 1.6 \times 10^{-5} m_\phi \sim 2.2 \times 10^8$ GeV and $\alpha < 10^{-8}$ to allow the Higgs to remain stable in most Hubble patches during reheating, and we also find the full two parameter joint constraints. We find a corresponding bound on the reheat temperature of $T_\text{reh} \lesssim 9.2 \times 10^9$ GeV. Additionally, de Sitter temperature fluctuations during inflation put a lower bound on inflaton-Higgs coupling by providing an effective mass for the Higgs, pushing back its hilltop during inflation. These additional constraints provide a lower bound on $\alpha$, while $\kappa$ must also be non-zero for the inflaton to decay efficiently.Comment: 12 pages in double column format, 12 figure