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Concept# Projective variety

Summary

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space \mathbb{P}^n over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of \mathbb{P}^n.
A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial.
If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring
:k[x_0, \ldots, x_n]/I
is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and t

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In the present thesis we study the geometry of the moduli spaces of Bradlow-Higgs triples on a smooth projective curve C. There is a family of stability conditions for triples that depends on a positive real parameter Ï. The moduli spaces of Ï-semistable triples of rank r and degree d vary with Ï. The phenomenon arising Ï from this is known as wall-crossing. In the first half of the thesis we will examine how the moduli spaces and their universal additive invariants change as Ï varies, for the case r = 2. In particular we will study the case of Ï very close to 0, for which the moduli space relates to the moduli space of stable Higgs bundles, and Ï very large, for which the moduli space is a relative Hilbert scheme of points for the family of spectral curves. Some of these results will be generalized to Bradlow-Higgs triples with poles. In the second half we will prove a formula relating the cohomology of the moduli spaces for small and odd degree and the perverse filtration on the cohomology of the moduli space of stable Higgs bundles. We will also partially generalize this result to the case of rank greater than 2.

The topic of this thesis is vanishing theorems in positive characteristic. In particular, we use "the covering trick of Ekedahl" to investigate the vanishing of $H^1(X, \mathcal{O}_X(-D))$ for a big and nef Weil divisor $D$ on a normal projective variety with $-K_X$ nef. In dimension two, we show that on a surface of log del Pezzo type over a perfect field of characteristic $p>5$ this vanishing holds. More generally, using techniques of the \emph{Minimal model program} we prove the Kawamata--Viehweg vanishing theorem in this setting. We also construct a counter-example in characteristic five, showing that our result is optimal. We discuss the relationship (due to Hacon--Witaszek) between this vanishing theorem and properties of threefold klt-singularities. We investigate if a similar relationship exists between threefold lc-singularities and a certain vanishing theorem for higher direct images of elliptic fibrations. This leads to a counter-example to a theorem of Koll'ar over the complex numbers, in every positive characteristic.

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