Ensuring sustainable food systems requires vastly reducing their environmental and health costs while making healthy and sustainable food affordable to all. One of the central problems of current food systems is that many of the costs of harmful foods are externalized, i.e., are not reflected in market prices. At the same time, the benefits of healthful foods are not appreciated. Due to externalities, sustainable and healthy food is often less affordable to consumers and less profitable for businesses than unsustainable and unhealthy food. Externalities and other market failures lead to unintended consequences for present and future generations, destroying nature and perpetuating social injustices such as underpay for workers, food insecurity, illness, premature death and other harms. We urgently need to address the fundamental causes of these problems. This chapter sets out the results of an analysis to determine the current cost of externalities in food systems and the potential impact of a shift in diets to more healthy and sustainable production and consumption patterns. The current externalities were estimated to be almost double (19.8 trillion USD) the current total global food consumption (9 trillion USD). These externalities accrue from 7 trillion USD (range 4–11) in environmental costs, 11 trillion USD (range 3–39) in costs to human life and 1 trillion USD (range 0.2–1.7) in economic costs. This means that food is roughly a third cheaper than it would be if these externalities were included. More studies are needed to quantify the costs and benefits of food systems that would support a global shift to more sustainable and healthy diets. However, the evidence presented in this chapter points to the urgent need for a system reset to account for these ‘hidden costs’ in food systems and calls for bold actions to redefine the incentives for producing and consuming healthier and more sustainable diets. The first step to correct for these ‘hidden costs’ is to redefine the value of food through true-cost accounting (TCA) so as to address externalities and other market failures. TCA reveals the true value of food by making the benefits of affordable and healthy food visible and revealing the costs of damage to the environment and human health 3.

The aim of this paper is to generalize the algorithm to compute jumping numbers on rational surfaces described in [AAD14] to varieties of dimension at least 3. Therefore, we introduce the notion of π-antieffective divisors, generalizing antinef divisors. Using these divisors, we present a way to find a small subset of the 'classical' candidate jumping numbers of an ideal, containing all the jumping numbers. Moreover, many of these numbers are automatically jumping numbers, and in many other cases, it can be easily checked.Proposition-Definition 2.5. Let a be an ideal sheaf on X. There exists an increasing sequence of rational numbers). The numbers λ i , i > 0, are called the jumping numbers of a.

We investigate some necessary and sufficient conditions for an exceptional divisor to contribute jumping numbers of an effective divisor on a variety of arbitrary dimension, inspired by the results for curves on surfaces by Smith and Thompson [ST07] and Tucker [Tuc10]. In particular, we construct an example of an exceptional divisor that is not contracted in the log canonical model, and does not contribute any jumping numbers.

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