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A Hessenberg variety is a subvariety of the flag variety parametrized by two maps: a Hessenberg function on $[n]$ and a linear map on $\C^n$. We study regular nilpotent Hessenberg varieties in Lie type A by focusing on the Hessenberg function $h=(n-1,n,\ldots,n)$. We first state a formula for the $f^w_{n,1}$ which generates the local defining ideal $J_{w,h}$ for any $w\in\Ss_n$. Second, we prove that there exists a convenient monomial order so that $\lead(J_{w,h})$ is squarefree. As a consequence, we conclude that each codimension-1 regular nilpotent Hessenberg variety is locally Frobenius split (in positive characteristic). Thesis Master of Science (MSc)

In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of n × n {n\times n} matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.

The goal of this paper is to construct a Frobenius splitting on $G/U$ via the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$, where $G$ is a simply connected semi-simple algebraic group defined over an algebraically closed field of characteristic $p> 3$, $U$ is the uniradical of a Borel subgroup of $G$, and $\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}}$ is the standard Poisson structure on $G/U$. We first study the Poisson geometry of $(G/U,\pi _{{{\scriptscriptstyle G}}/{{\scriptscriptstyle U}}})$. Then we develop a general theory for Frobenius splittings on $\mathbb{T}$-Poisson varieties, where $\mathbb{T}$ is an algebraic torus. In particular, we prove that compatibly split subvarieties of Frobenius splittings constructed in this way must be $\mathbb{T}$-Poisson subvarieties. Lastly, we apply our general theory to construct a Frobenius splitting on $G/U$.

We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field $\bK$ of characteristic $\neq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind)scheme structure, its projective coordinate ring has a $\Z$-model, and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. {\bf 371} no.2 (2018)]) when $\mathsf{char} \, \bK =0$ or $\gg 0$, and the higher cohomology vanishing of their nef line bundles in arbitrary characteristic $\neq 2$. Some particular cases of these results play crucial roles in our proof [K, arXiv:1805.01718] of a conjecture by Lam-Li-Mihalcea-Shimozono [J. Algebra {\bf 513} (2018)] that describes an isomorphism between affine and quantum $K$-groups of a flag manifold., 64pages, v4: added Appendix A, v5: corrected extremal condition of Theorem C, and added Appendix B, v6 and v7: minor improvements

Generalizing the notion of Frobenius-splitting, we prove that every finite height Calabi–Yau variety defined over an algebraically closed field of positive characteristic can be lifted to the ring of Witt vectors of length two.

We prove that for a simply laced group, the closure of the Borel conjugacy class of any nilpotent element of height 2 in its conjugacy class is normal and admits a rational resolution. We extend this, using Frobenius splitting techniques, to the closure in the whole Lie algebra if either the group has type A or the element has rank 2.

We explain that the Plücker relations provide the defining equations of the thick flag manifold associated to a Kac–Moody algebra. This naturally transplants the result of Kumar–Mathieu–Schwede about the Frobenius splitting of thin flag varieties to the thick case. As a consequence, we provide a description of the space of global sections of a line bundle of a thick Schubert variety as conjectured in Kashiwara–Shimozono [13]. This also yields the existence of a compatible basis of thick Demazure modules and the projective normality of the thick Schubert varieties.

In this paper, we study the Frobenius split property of fibrations over curves via the relative Frobenius morphism. In particular, we can classify Frobenius split Abelian fiber spaces over curves which has no wild fibers in terms of the degree of the top higher direct image R n − 1 π ⁎ O X , the multiplicities of multiple fibers, non-ordinary points, and the characteristic of the base field.

I provide a construction of intrinsic weakly Ulrich bundles of large rank on any smooth complete surface in ${\bf P}^3$ over fields of characteristic $p>0$ and also for some classes of surfaces of general type in ${\bf P}^n$. I also construct intrinsic weakly Ulrich bundles on any Frobenius split variety of dimension at most three. The bundles constructed here are in fact ACM and weakly Ulrich bundles and so I call them almost Ulrich bundles. Frobenius split varieties in dimension three include as special cases: (1) smooth hypersurfaces in ${\bf P}^4$ of degree at most four, (2) Frobenius split, smooth quintics in ${\bf P}^4$ (3) Frobenius split Calabi-Yau varieties of dimension at most three (4) Frobenius split (i.e ordinary) abelian varieties of dimension at most three., 7 pages; To appear in Advances in Math

In this paper we consider Artinian modules over power series rings endowed with a Frobenius map. We describe a method for finding the set of all prime annihilators of submodules which are preserved by the given Frobenius map and on which the Frobenius map is not nilpotent. This extends the algorithm by Karl Schwede and the first author, which solved this problem for submodules of the injective hull of the residue field. The Matlis dual of this problem asks for the radical annihilators of quotients of free modules by submodules preserved by a given Frobenius near-splitting, and the same method solves this dual problem in the F-finite case. © 2013 Elsevier B.V.

Recently E. Feigin introduced the $\mathbb G_a^N$-degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann that the degenerate flag varieties in types $A_n$ and $C_n$ are Frobenius split. In this paper we construct an associated degeneration of homogeneous coordinate rings of classical flag varieties in all types and show that these rings are Frobenius split in most types. It follows that the degenerate flag varieties of types $A_n$, $C_n$ and $G_2$ are Frobenius split. In particular we obtain an alternate proof of splitting in types $A_n$ and $C_n$; the case $G_2$ was not previously known. We also give a representation-theoretic condition on PBW-graded versions of Weyl modules which is equivalent to the existence of a Frobenius splitting of the classical flag variety that maximally compatibly splits the identity., Comment: Bug fixes; final version. To appear in Selecta Mathematica

Let $T$ be a maximal torus of $\operatorname{PSL}(n,\mathbb{C})$ . For $n\geq 4$ , we construct a smooth compactification of $\operatorname{PSL}(n,\mathbb{C})/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{\operatorname{PSL}(n,\mathbb{C})}$ for a suitable choice of $T$ -linearized ample line bundle on $\overline{\operatorname{PSL}(n,\mathbb{C})}$ . We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of $\operatorname{PSL}(n,\mathbb{C})/T$ is $\operatorname{PSL}(n,\mathbb{C})$ itself.

Let G be a reductive group over an algebraically closed field of characteristic p>0. We study properties of embeddings of spherical homogeneous G-spaces. We look at Frobenius splittings, canonical or by a (p-1)-th power, compatible with certain subvarieties. We also look at cohomology vanishing and show the existence of rational G-equivariant resolutions by toroidal embeddings. We show that the class of homogeneous spaces for which our results hold contains the symmetric homogeneous spaces in characteristic not 2 and is closed under parabolic induction., Expanded and corrected version

Let $X^v_w$ be a Richardson variety in the full flag variety $X$ associated to a symmetrizable Kac-Moody group $G$. Recall that $X^v_w$ is the intersection of the finite dimensional Schubert variety $X_w$ with the finite codimensional opposite Schubert variety $X^v$. We give an explicit $\bQ$-divisor $\Delta$ on $X^v_w$ and prove that the pair $(X^v_w, \Delta)$ has Kawamata log terminal singularities. In fact, $-K_{X^v_w} - \Delta$ is ample, which additionally proves that $(X^v_w, \Delta)$ is log Fano. We first give a proof of our result in the finite case (i.e., in the case when $G$ is a finite dimensional semisimple group) by a careful analysis of an explicit resolution of singularities of $X^v_w$ (similar to the BSDH resolution of the Schubert varieties). In the general Kac-Moody case, in the absence of an explicit resolution of $X^v_w$ as above, we give a proof that relies on the Frobenius splitting methods. In particular, we use Mathieu's result asserting that the Richardson varieties are Frobenius split, and combine it with a result of N. Hara and K.-I. Watanabe relating Frobenius splittings with log canonical singularities., Comment: 15 pages, improved exposition and explanation. To appear in the International Mathematics Research Notices

Let $R$ be a ring of prime characteristic $p$, and let $F^e_* R$ denote $R$ viewed as an $R$-module via the $e$th iterated Frobenius map. Given a surjective map $\phi : F^e_* R \to R$ (for example a Frobenius splitting), we exhibit an algorithm which produces all the $\phi$-compatible ideals. We also explore a variant of this algorithm under the hypothesis that $\phi$ is not necessarily a Frobenius splitting (or even surjective). This algorithm, and the original, have been implemented in Macaulay2., Comment: 15 pages, many statements clarified and numerous other substantial improvements to the exposition (thanks to the referees). To appear in the Journal of Symbolic Computation

We study the singularities of the irreducible components of the Springer fiber over a nilpotent element N with N 2 = 0 in a Lie algebra of type A or D (the so-called two columns case). We use Frobenius splitting techniques to prove that these irreducible components are normal, Cohen–Macaulay, and have rational singularities.

Lakshmibai, Mehta and Parameswaran (LMP) introduced the notion of maximal multiplicity vanishing in Frobenius splitting. In this paper we define the algebraic analogue of this concept and construct a Frobenius splitting vanishing with maximal multiplicity on the diagonal of the full flag variety. Our splitting induces a diagonal Frobenius splitting of maximal multiplicity for a special class of smooth Schubert varieties first considered by Kempf. Consequences are Frobenius splitting of tangent bundles, of blow-ups along the diagonal in flag varieties along with the LMP and Wahl conjectures in positive characteristic for the special linear group., Comment: Revised according to referee suggestions. To appear in Annales de l'Institut Fourier

Let G be a semisimple algebraic group over an algebraically closed field of positive characteristic. In this note, we show that an irreducible closed subvariety of the flag variety of G is compatibly split by the unique canonical Frobenius splitting if and only if it is a Richardson variety, i.e. an intersection of a Schubert and an opposite Schubert variety., After obtaining this result, I was informed that Allen Knutson has also recently obtained a similar result. This version fixes an error in Lemma 2.7. To appear in J. Algebra

We construct a full, strongly exceptional collection of line bundles on the variety X that is the blow up of the projectivization of the vector bundle O_{Pn−1} ⊕ O_{Pn−1}(b_1) along a linear space of dimension n − 2, where b_1 is a non-negative integer.

Let $X$ be an equivariant embedding of a connected reductive group $G$ over an algebraically closed field $k$ of positive characteristic. Let $B$ denote a Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In the case where $X$ is the wonderful compactification of a group of adjoint type, the $G$-Schubert varieties are the closures of Lusztig's $G$-stable pieces. We prove that $X$ admits a Frobenius splitting which is compatible with all $G$-Schubert varieties. Moreover, when $X$ is smooth, projective and toroidal, then any $G$-Schubert variety in $X$ admits a stable Frobenius splitting along an ample divisors. Although this indicates that $G$-Schubert varieties have nice singularities we present an example of a non-normal $G$-Schubert variety in the wonderful compactification of a group of type $G_2$. Finally we also extend the Frobenius splitting results to the more general class of $\mathcal R$-Schubert varieties., Final version, 44 pages

We study the structure of Frobenius splittings (and generalizations thereof) induced on compatible subvarieties $W \subseteq X$. In particular, if the compatible splitting comes from a compatible splitting of a divisor on some birational model $E \subseteq X' \to X$ (ie, this is a log canonical center), then we show that the divisor corresponding to the splitting on $W$ is bounded below by the divisorial part of the different as studied by Kawamata, Shokurov, Ambro and others. We also show that difference between the divisor associated to the splitting and the divisorial part of the different is largely governed by the (non-)Frobenius splitting of fibers of $E \to W$. In doing this analysis, we recover an $F$-canonical bundle formula by reinterpretting techniques common in the theory of Frobenius splittings., 27 pages, numerous typos corrected, the exposition is also improved

This paper establishes uniform bounds in characteristic p p rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the F-signature function. From this we establish that the F-signature function is lower semi-continuous. Lower semi-continuity of the F-signature of a pair is also established. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, which was originally proven by Ilya Smirnov.

We describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This "bipartite Zelevinsky map" restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type A quivers of arbitrary orientation, we give the same result up to some factors of general linear groups. These identifications allow us to recover results of Bobinski and Zwara; namely we see that orbit closures of type A quivers are normal, Cohen-Macaulay, and have rational singularities. We also see that each representation space of a type A quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split., 24 pages, comments welcome. v2: Section 3.2 new, Theorem 4.20 improved, references added

Let $T$ be a maximal torus of ${\rm PSL}(n, \mathbb C)$. For $n\,\geq\, 4$, we construct a smooth compactification of ${\rm PSL}(n, \mathbb C)/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{{\rm PSL}(n, \mathbb C)}$ for a suitable choice of $T$--linearized ample line bundle on $\overline{{\rm PSL}(n, \mathbb C)}$. We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of ${\rm PSL}(n, \mathbb C)/T$ is ${\rm PSL}(n, \mathbb C)$ itself.

We show that locally acyclic cluster algebras have (at worst) canonical singularities. In fact, we prove that locally acyclic cluster algebras of positive characteristic are strongly F-regular. In addition, we show that upper cluster algebras are always Frobenius split by a canonically defined splitting, and that they have a free canonical module of rank one. We also give examples to show that not all upper cluster algebras are F-regular if the local acyclicity is dropped., 24 pages