Coherent Systems and Brill–Noether Theory

Abstract

Suplico a vuesa merced, señor don Quijote, que mire bien y especule con cien ojos lo que hay allá dentro: quizá habrá cosas que las ponga yo en el libro de mis Transformaciones (El ingenioso hidalgo don Quijote de la Mancha, Book 2, Chapter XXII) I beg you, don Quixote sir: look carefully, inspect with a hundred eyes what you see down there. Who knows, maybe you will find something that I can put in my book on Transformations.

Hodge polynomials and birational types of moduli spaces of coherent systems on elliptic curves

“…1 − uv in accordance with Proposition 3.1 (2). For i = 0 and d even, it can easily be checked that the formula of Theorem 6.7 agrees with Theorem 5.9(a).…”

COHERENT SYSTEMS OF GENUS 0 II: EXISTENCE RESULTS FOR k ≥ 3

“…Essentially a critical value α c is a value of α at which the α-stability condition for a coherent system (E, V ) can change as α passes through the value α c , while the corresponding critical data sets describe the way in which this change takes place. For convenience we write G(α As in [1] (and putting g = 0), we define for any critical data set (6)…”

“…We now start on the computations of C 12 and C 21 , where we continue to assume that 0 < k < n. With the notation of sections 1 and 2, note that, by [1,Lemma 6.5], the flip loci at any critical value can be constructed using only those critical data sets for which there exist (E 1 , V 1 ) and (E 2 , V 2 ) which are both α-stable either for α = α − c or for α = α + c . Since we prefer to have purely numerical conditions on our critical data sets, we shall say that…”

“…A coherent system is called α-stable (α-semistable) if µ α (F, W ) < µ α (E, V ) (µ α (F, W ) ≤ µ α (E, V )) for every proper coherent subsystem (F, W ) of (E, V ). According to general theory (see, for example, [1]), there exists a moduli space of α-stable coherent systems of type (n, d, k), which we denote by G(α; n, d, k).…”

“…The first is a development of an argument used in [2], whereby the existence problem for small positive values of α is reduced to a problem in projective geometry, which we solve completely. The second method is completely different from those of [2], depending on an analysis of the "flips" introduced in [1]. The advantage of this approach is that it makes it possible to translate results from one value of α to another.…”