Abstract. We study the projective normality of a minimal surface X which is a ramified double covering over a rational surface S with dim | − K S | ≥ 1. In particular Horikawa surfaces, the minimal surfaces of general type with K 2 X = 2p g (X)−4, are of this type, up to resolution of singularities. Let π be the covering map from X to S . We show that the Z 2 -invariant adjoint divisors K X + rπ * A are normally generated, where the integer r ≥ 3 and A is an ample divisor on S .

For a smooth quasi-projective surface X and an integer n ≥ 3, we show that the universal family Z n over the Hilbert scheme Hilb n (X) of n points has non Q-Gorenstein, rational singularities, and that the Samuel multiplicity µ at a closed point on Z n can be computed in terms of the dimension of the socle. We also show that µ ≤ n.

Let [Formula: see text] be a non-degenerate normal projective variety of codimension [Formula: see text] and degree [Formula: see text] with isolated [Formula: see text]-Gorenstein singularities. We prove that the Castelnuovo–Mumford regularity [Formula: see text], as predicted by the Eisenbud–Goto regularity conjecture. Such a bound fails for general projective varieties by a recent result of McCullough–Peeva. The main techniques are Noma’s classification of non-degenerate projective varieties and Nadel vanishing for multiplier ideals. We also classify the extremal and the next to extremal cases.

Abstract. We study the singularities of the secant variety Σ(X, L) associated to a smooth variety X embedded by a sufficiently positive adjoint line bundle L. We show that Σ(X, L) is always Du Bois singular. Examples of secant varieties with worse singularities when L has weak positivity are provided. We also give a necessary and sufficient condition for Σ(X, L) to have rational singularities.

The Hitchin morphism is a map from the moduli space of Higgs bundles M X to the Hitchin base B X , where X is a smooth projective variety. When X has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme A X of B X , which is called the space of spectral data. They proved that the Hitchin morphism factors through A X and conjectured that A X is the image of the Hitchin morphism. We prove that when X is a smooth projective surface, this conjecture is true for vector bundles. Moreover, we show that A X (for any dimension) is invariant under any proper birational morphism, and apply the result to study A X for ruled surfaces.