Arc Schemes Research Articles

Foreword to the special issue of the Australian Journal of Grape and Wine Research highlighting research of the Australian Research Council Training Centre for Innovative Wine Production

This special issue of the Australian Journal of Grape and Wine Research is a compilation of review and research articles from staff, students, industry partners and collaborators of the Australian Research Council Training Centre for Innovative Wine Production (ARC TC IWP). The Centre, a partnership between The University of Adelaide, Charles Sturt University and wine industry organisations, was initially funded for 3 years in the first round (2013) of the ARC scheme. The Centre's key objective was to develop an integrated, whole-of-production-chain approach to manage flavour and alcohol content in Australian wine production. Examples of key findings were one of the first Australian studies of the impact of water addition to fermentations to manage wine alcohol content. The work nicely pre-empted the change in the Australia New Zealand Food Standards Code that permitted this practice in Australia and helped frame recommendations to winemakers and researchers about potential effects on wine. In another study, the most comprehensive genetic and phenotypic analysis of the yeast, Lachancea thermotolerans, resulted in the identification of strains with strong potential as bioacidification agents with reduced ethanol yield. Isolates characterised in the work will ideally become commercially available in the near future and will be well suited to fermenting overripe grapes. Of course the other significant output was a cohort of highly skilled wine researchers with experience in working at the research–industry interface. Many have remained in education, technical or research roles linked to the sector through partner organisations or, in fact, the current Centre. More details about the first iteration of the Centre and further project outcomes can be found at www.arcwinecentre.org.au/history-2014-2017/.

Motivic volumes of fibers of tropicalization

Let T be an algebraic torus over an algebraically closed field, let X be a smooth closed subvariety of a T -toric variety such that U = X \cap T is not empty, and let \mathscr{L}(X) be the arc scheme of X . We consider a tropicalization map on \mathscr{L}(X) \setminus \mathscr{L}(X \setminus U) , the set of arcs of X that do not factor through X \setminus U . We show that each fiber of this tropicalization map is a constructible subset of \mathscr{L}(X) and therefore has a motivic volume. We prove that if U has a compactification with simple normal crossing boundary, then the generating function for these motivic volumes is rational, and we express this rational function in terms of certain lattice maps constructed in Hacking, Keel, and Tevelev's theory of geometric tropicalization. We explain how this result, in particular, gives a formula for Denef and Loeser's motivic zeta function of a polynomial. To further understand this formula, we also determine precisely which lattice maps arise in the construction of geometric tropicalization.

On the nilpotent functions at a non-degenerate arc

Let k be an arbitrary field and X be a reduced k-scheme of finite type. Let $$\gamma $$ be an arc on X, not entirely contained in the non-smooth locus of X. We show that the nilpotents in the local ring at $$\gamma $$ vanish in the completion. Along the way, we also obtain informations on the connection between torsion differential forms and nilpotent functions on arc schemes in positive characteristic.

Finite formal model of toric singularities

We study the formal neighborhoods at rational non-degenerate arcs of the arc scheme associated with a toric variety. The first main result of this article shows that these formal neighborhoods are generically constant on each Nash component of the variety. Furthermore, using our previous work, we attach to every such formal neighborhood, and in fact to every toric valuation, a minimal formal model (in the class of stable isomorphisms) which can be interpreted as a measure of the singularities of the base-variety. As a second main statement, for a large class of toric valuations, we compute the dimension and the embedding dimension of such minimal formal models, and we relate the latter to the Mather discrepancy. The class includes the strongly essential valuations, that is to say those the center of which is a divisor in the exceptional locus of every resolution of singularities of the variety. We also obtain a similar result for monomial curves.

The minimal formal models of curve singularities

Let [Formula: see text] be a field. We introduce a new geometric invariant, namely the minimal formal models, associated with every curve singularity (defined over [Formula: see text]). This is a noetherian affine adic formal [Formula: see text]-scheme, defined by using the formal neighborhood in the associated arc scheme of a primitive [Formula: see text]-parametrization. For the plane curve [Formula: see text]-singularity, we show that this invariant is [Formula: see text]. We also obtain information on the minimal formal model of the so-called generalized cusp. We introduce various questions in the direction of the study of these minimal formal models with respect to singularity theory. Our results provide the first positive elements of answer. As a direct application of the former results, we prove that, in general, the isomorphisms satisfying the Drinfeld–Grinberg–Kazhdan theorem on the structure of the formal neighborhoods of arc schemes at non-degenerate arcs do not come from the jet levels. In some sense, this shows that the Drinfeld–Grinberg–Kazhdan theorem is not a formal consequence of the Denef–Loeser fibration lemma.

A remark on Berger’s conjecture, Kolchin’s theorem, and arc schemes

Let k be a field of characteristic zero. Let V be a k-scheme of finite type, i.e., a k-variety, which is integral. We prove that if the associated arc scheme \({\mathcal{L}_{\infty}(V)}\) is reduced, then the \({\mathcal{O}_{V}}\)-Module \({\Omega_{V/k}^{1}}\) is torsion-free. Then if the k-variety V is assumed to be locally a complete intersection (lci), we deduce that the k-variety V is normal. We also obtain the following consequence: for every class \({\mathfrak{C}}\) of integral k-curves which satisfies the Berger conjecture, and for every \({\mathscr{C} \in \mathfrak{C}}\), the k-curve \({\mathscr{C}}\) is smooth if and only if \({\mathcal{L}(\mathscr{C})}\) is reduced.

Nilpotency in arc scheme of plane curves

ABSTRACTLet k be a field of characteristic 0. Let be a non-constant reduced polynomial. In this article we study the differential ideal of the differential ring . We introduce the S-algorithm which allows, for every differential polynomial , to test if it belongs or not to the differential ideal . Another consequence of this study is to establish a precise characterization of the reducedness of the arc scheme associated with a plane curve X, defined over k, in terms of the geometry of X.

THE DRINFELD–GRINBERG–KAZHDAN THEOREM IS FALSE FOR SINGULAR ARCS

In this note, we prove that the Drinfeld–Grinberg–Kazhdan theorem on the structure of formal neighborhoods of arc schemes at a nonsingular arc does not extend to the case of singular arcs.

The demise of the ARC journal ranking scheme: an ex post analysis of the accounting and finance journals

AbstractThe recent abolition of the ARC journal ranking scheme is indicative of some problematical features of journal ranking in general and the ARC scheme in particular. An alternative citation‐based ranking scheme is applied to the accounting and finance journals to highlight some loopholes in the abandoned ARC scheme and provide some suggestions for how to proceed with ERA 2012. By re‐ranking journals according to their citation indices, it is demonstrated that the ARC ranking placed a large number of journals where they do not belong. As a result, the ARC scheme induced some adverse behavioural changes with respect to preferred publication outlets.

Women in Chemistry in Australia: From a Slow Start to a More Promising Future

Margaret M. Sheil shares her views on the role and growth of women chemists in Australia. Women have been rare in chemistry and even rarer in the ranks of leadership of academic chemistry and in the main professional association, the Royal Australian Chemical Institute (RACI). Prior to 2009 when Ada Yonath was awarded the Nobel Prize for Chemistry, the last women chemist to be awarded a Nobel Prize was Dorothy Hodgson in 1964. The author goes through several research works to know more about women chemists. She comes across many women chemists but feels sad that they were not given their due credit. She also researched that the highest number of female applicants is found in physical and structural chemistry and the lowest in inorganic chemistry. To help address the small number of successful women in this most prestigious ARC scheme, the Australian Laureate Fellowships, the ARC has added two additional fellowships in 2011.

Greenberg Approximation and the Geometry of Arc Spaces

We study the differential properties of generalized arc schemes and geometric versions of Kolchin's Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves.

Toric arc schemes and quantum cohomology of toric varieties

We describe the quantum cohomology ring of a toric Fano variety $X$ in terms of the usual topological cohomology ring for an auxiliary infinite-dimensional scheme. This scheme is a part of an algebro-geometric model for the universal cover of the space of free loops with values in $X$.