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Let$X$and$Y$be smooth varieties of dimensions$n$−1 and$n$over an arbitrary algebraically closed field,$f: X→Y$a finite map that is birational onto its image. Suppose that$f$is curvilinear; that is, for all$xεX$, the Jacobian ϱ$f(x)$has rank at least$n$−2. For$r$≥1, consider the subscheme$N$_{$r$}of$Y$defined by the ($r$−1)th Fitting ideal of the $$\mathcal{O}_Y $$ -module $$f_ * \mathcal{O}_X $$ , and set$M$_{$r$}∶=$f$^{−1}$N$_{$r$}. In this setting—in fact, in a more general setting—we prove the following statements, which show that$M$_{$r$}and$N$_{$r$}behave like reasonable schemes of source and target$r$-fold points of$f$.

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In this paper, we consider the L p dual Minkowski problem by geometric variational method. Using anisotropic Gauss–Kronecker curvature flows, we establish the existence of smooth solutions of the L p dual Minkowski problem when pq ≥ 0 and the given data is even. If f ≡ 1, we show under some restrictions on p and q that the only even, smooth, uniformly convex solution is the unit ball.

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