Global aspects of the geometry of surfaces

These notes (prepared for the author's lectures at the Cracow Summer School on Linear Systems organized by S. Mueller-Stach and T. Szemberg, held March 23-27, 2009 at the Pedagogical University of Cracow under the sponsorship of the Deutsche Forschungsgemeinschaft) present a number of open problems on the theory of smooth projective algebraic surfaces, and put into historical context recent work on a range of topics, including Mori dream spaces and the finite generation of the Cox ring, Seshadri constants, and the resurgence of homogeneous ideals and the problem of which ordinary powers of homogeneous ideals contain given symbolic powers of those ideals. These notes include many exercises, with solutions.Comment: 23 pages; exercises are included, with solutions (revision includes positive characteristic example pointed out by J. Koll\'ar and brought to my attention by B. Totaro of surface with curves C for which C^2 is arbitrarily negative

Anticanonical Rational Surfaces

A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, treated in separate papers, include questions involving: points in good position, birational models of rational surfaces in projective space, and resolutions for ideals of fat point subschemes of $P^2$.Comment: 14 pp. The preprint itself is not contained in the Duke archive; plainTeX textfile and dvi versions of this preprint can instead be obtained via the author's www site, http://www.math.unl.edu/~bharbour/ . Comments and requests can be directed to [email protected]

On Nagata's Conjecture

Modifying an approach of J. Roe, this paper gives an improved lower bound on the degrees d such that for general points p1,...,pn in P2 and m > 0 there is a plane curve of degree d vanishing at each point pi with multiplicity at least m. In certain cases, for m not too large compared with n, the new bound implies a bound conjectured by Nagata.Comment: 6 page

Configuration types and cubic surfaces

This paper is a sequel to the paper \cite{refGH}. We relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type. Configuration types arise when one classifies the Hilbert functions and graded Betti numbers for fat point subschemes supported at $n\le8$ essentially distinct points of the projective plane. Each type gives rise to a surface $X$ obtained by blowing up the points. We classify those types such that $n=6$ and $-K_X$ is nef. The surfaces obtained are precisely the desingularizations of the normal cubic surfaces. By classifying configuration types we recover in all characteristics the classification of normal cubic surfaces, which is well-known in characteristic 0 \cite{refBW}. As an application of our classification of configuration types, we obtain a numerical procedure for determining the Hilbert function and graded Betti numbers for the ideal of any fat point subscheme $Z=m_1p_1+...+m_6p_6$ such that the points $p_i$ are essentially distinct and $-K_X$ is nef, given only the configuration type of the points $p_1,...,p_6$ and the coefficients $m_i$.Comment: 14 pages, final versio