Related papers: Classification of Horikawa surfaces with T-singula…

We identify the stable surfaces around the stable limit of the examples of Y. Lee and J. Park [LP07], and H. Park, J. Park and D. Shin [PPS09] using the explicit 3-fold Mori theory in [HTU13]. These surfaces belong to the…

Let A be an indecomposable principally polarized abelian variety of dimension g . Third order theta functions embed A in a projective space P(V_3), while second order theta functions embed the Kummer variety K=A/<-1> in a projective space…

A smooth, projective surface $S$ is called a $\emph{standard isotrivial fibration}$ if there exists a finite group $G$ which acts faithfully on two smooth projective curves $C$ and $F$ so that $S$ is isomorphic to the minimal…

We show that Hilbert schemes of points on supersingular Enriques surface in characteristic 2 are simply connected, symplectic varieties but are not irreducible symplectic as the hodge number $h^{2,0} > 1$, even though a supersingular…

We describe Baily-Borel, toroidal, and geometric -- using the KSBA stable pairs -- compactifications of some moduli spaces of K3 surfaces with a nonsymplectic automorphism of order $3$ and $4$ for which the fixed locus of the automorphism…

We consider singular Q-acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is C^1- or C*-ruled or the surface is up to isomorphism one of two…

We prove that Godeaux--Reid surfaces with torsion group Z/3 have topological fundamental group Z/3. For this purpose, we describe degenerations to stable KSBA surfaces with one 1/4(1,1) singularity, whose minimal resolution are elliptic…

In the closed, non-Haken, hyperbolic class of examples generated by (2p,q) Dehn fillings of Figure 8 knot space, the geometrically incompressible one-sided surfaces are identified by the filling ratio p/q and determined to be unique in all…

In this paper the authors consider a certain toroidal compactification of the moduli space of degenerations of (1,p)-polarized abelian surfaces with (canonical) level structure. Using Hodge theory we give a proof that a degenerate abelian…

We show that the normal points of a cubic hypersurface in projective space have canonical singularities unless the hypersurface is an iterated cone over an elliptic curve. As an application, we give a simple linear algebraic description of…

We classify smooth surfaces whose higher cohomologies of i-forms for all i vanish. We show that if such a surface is not affine, then it has essentially two possibilities.

We prove that the affine cone over a general primitively polarised K3 surface of genus g is smoothable if and only if g\le 10 or g=12. We also give several examples of singularities with special behaviour, such as surfaces whose affine cone…

We explicitly describe the KSBA/Hacking compactification of a moduli space of log surfaces of Picard rank 2. The space parametrizes log pairs $(S, D)$ where $S$ is a degeneration of $\mathbb{P}^1 \times \mathbb{P}^1$ and $D \subset S$ is a…

We prove that every irreducible component of the coarse Koll\'ar-Shepherd-Barron and Alexeev (KSBA) moduli space of stable log Calabi--Yau surfaces admits a finite cover by a projective toric variety. This verifies a conjecture of…

One of the simplest examples of a smooth, non degenerate surface in P^4 is the quintic elliptic scroll. It can be constructed from an elliptic normal curve E by joining every point on E with the translation of this point by a non-zero…

Using results by Donaldson and Auroux on pseudo-holomorphic curves as well as Duval's rational convexity construction, the paper investigates the existence of smooth Lagrangian surfaces representing 2-dimensional homology classes in complex…

Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type with nef canonical divisor and canonical singularities, whose image $Y$ under the canonical map…

We study minimal complex surfaces S of general type with q(S)=q and p_g(S)=2q-3, q>= 5. We give a complete classification in case that S has a fibration onto a curve of genus >=2. For these surfaces K^2=8\chi. In general we prove that…

We study the cohomology groups $H^1(X,\Theta_X(-mK_X))$, for $m\geq1$, where $X$ is a smooth minimal complex surface of general type, $\Theta_X$ its holomorphic tangent bundle, and $K_X$ its canonical divisor. One of the main results is a…

We classify curves in the moduli space of curves that are both Shimura- and Teichmueller curves: Except for the moduli space of genus one curves there is only a single such curve. We start with a Hodge-theoretic description of Shimura…