Related papers: Smoothing Toric Fano Surfaces Using the Gross-Sieb…
Let $V$ be an affine algebraic variety, and let $p\in V$ be a singular point. For a regular function $g$ on $V$ such that $g(p)=0$ and for a positive integer $n$, we consider the cyclic covering $\phi_n\: V_n \to V$ of degree $n$ branched…
A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2\pi\Z^n$, is a smooth projective toric variety completely…
In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L. Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations of smooth Fano threefolds by projections was introduced; using Mirror Symmetry these…
Algebraic varieties are the geometric shapes defined by systems of polynomial equations; they are ubiquitous across mathematics and science. Amongst these algebraic varieties are Q-Fano varieties: positively curved shapes which have…
We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that…
A generalized semitoric system F:=(J,H): M --> R^2 on a symplectic 4-manifold is an integrable system whose essential properties are that F is a proper map, its set of regular values is connected, J generates an S^1-action and is not…
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of…
The Milnor fibre of a $\mathbb{Q}$-Gorenstein smoothing of a Wahl singularity is a rational homology ball $B_{p,q}$. For a canonically polarised surface of general type $X$, it is known that there are bounds on the number $p$ for which…
The construction of parametric curve and surface plays important role in computer aided geometric design (CAGD), computer aided design (CAD), and geometric modeling. In this paper, we define a new kind of blending functions associated with…
In this article we explore compactifications of cluster varieties of finite type in complex dimension two. Cluster varieties can be viewed as the spec of a ring generated by theta functions and a compactification of such varieties can be…
We prove that the Landau--Ginzburg superpotential of del Pezzo surfaces can be realized as a limit of their hyperK\"ahler rotation toward the large complex structure limit point. As a corollary, we compute the limit of the complex affine…
Let X be a surface with quotient singularities which admits a smoothing to the plane. We prove that X is a deformation of a weighted projective plane P(a^2,b^2,c^2), where a,b,c is a solution of the Markov equation a^2+b^2+c^2=3abc. We also…
We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it…
Algebraically simply connected surfaces of general type with p_g=q=0 and 1\le K^2\le 4 in positive characteristic (with one exception in K^2=4) are presented by using a Q-Gorenstein smoothing of two-dimensional toric singularities, a…
We generalized the construction of deformations of affine toric varieties of K. Altmann and our previous construction of deformations of weak Fano toric varieties to the case of arbitrary toric varieties by introducing the notion of…
Recent work ([18], [1]) has produced a complete list of weighted homogeneous surface singularities admitting smoothings whose Milnor fibre has only trivial rational homology (a "rational homology disk"). Though these special singularities…
We study (smooth, complex) Fano 4-folds X with Picard number rho(X)>6. We show that if rho(X)>9, then X is a product of del Pezzo surfaces, thus improving recent results by the author and by the author and S.A. Secci; the statement is now…
We study the Landau-Ginzburg mirror of toric/non-toric blowups of (possibly non-Fano) toric surfaces arising from SYZ mirror symmetry. Through the framework of tropical geometry, we provide an effective method for identifying the precise…
We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.
Motivated by an application to LDPC (low density parity check) algebraic geometry codes described by Voloch and Zarzar, we describe a computational procedure for establishing an upper bound on the arithmetic or geometric Picard number of a…